Metamath Proof Explorer - mmtheorems53

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Statement List for Metamath Proof Explorer - 5201-5300 - Page 53 of 58

Type	Label	Description
Statement
Theorem	projlem9 5201	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Real closure of the infimum of norms. Used by projlem11 5203 projlem12 5204 projlem13 5205 projlem15 5207.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    ⇒   ⊢ sup(S, ℝ, < ) ∈ ℝ
Theorem	projlem10 5202	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). A member of the infimum set. Used by projlem12 5204.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    ⇒   ⊢ (B ∈ H → -(norm ‘(B −v A)) ∈ S)
Theorem	projlem11 5203	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). R is the infimum of the set of norms. Show it is real. Used by projlem12 5204 projlem13 5205 projlem14 5206 projlem15 5207 projlem18 5210 projlem19 5211 projlem26 5218 projlem28 5220 projlem31 5223.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    ⇒   ⊢ R ∈ ℝ
Theorem	projlem12 5204	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). The infimum is less than any norm in the set of norms. Used by projlem14 5206 projlem18 5210 projlem31 5223.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    ⇒   ⊢ (B ∈ H → R ≤ (norm ‘(B −v A)))
Theorem	projlem13 5205	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). The infimum of the set of norms is nonnegative. Used by projlem18 5210 projlem19 5211 projlem28 5220.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    ⇒   ⊢ 0 ≤ R
Theorem	projlem14 5206	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 5208.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ C ∈ ℕ    &   ⊢ B ∈ H    ⇒   ⊢ (R − (1 / C)) < (norm ‘(B −v A))
Theorem	projlem15 5207	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 5208.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ C ∈ ℕ    ⇒   ⊢ ∃z ∈ H (norm ‘(z −v A)) < (R + (1 / C))
Theorem	projlem16 5208	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Existence of a vector sequence member, used to show (via Axiom of Choice) the vector sequence existence. Used by projlem17 5209.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ C ∈ ℕ    ⇒   ⊢ ∃z ∈ H ((R − (1 / C)) < (norm ‘(z −v A)) ∧ (norm ‘(z −v A)) < (R + (1 / C)))
Theorem	projlem17 5209	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). This uses the Axiom of Choice to show the existence of a vector sequence satisfying the assumption of Lemma 3.6 of [Beran] p. 100: "Let {yn } be a sequence of W such that i0 - 1/n < ||x0 - yn || < i0 + 1/n." Here, H corresponds to "W"; f:ℕ–→H to "{yn }"; w to "n"; R to "i0 "; and (norm ‘(A −v (f ‘w))) to "||x0 - yn ||". Used by projlem 5224.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    ⇒   ⊢ ∃f(f:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((f ‘w) −v A)) ∧ (norm ‘((f ‘w) −v A)) < (R + (1 / w))))
Theorem	projlem18 5210	Part of Lemma 3.6 of [Beran] p. 101, top. Used by projlem19 5211.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ B ∈ H    &   ⊢ C ∈ H    ⇒   ⊢ (4 · (R↑2)) ≤ ((norm ‘((B +v C) −v (2 ·s A)))↑2)
Theorem	projlem19 5211	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem20 5212.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ B ∈ H    &   ⊢ C ∈ H    &   ⊢ D ∈ ℕ    &   ⊢ G ∈ ℕ    &   ⊢ N ∈ ℕ    ⇒   ⊢ (((norm ‘(B −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(B −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N))))
Theorem	projlem20 5212	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem27 5219.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ N ∈ ℕ    ⇒   ⊢ (((B ∈ H ∧ C ∈ H) ∧ (D ∈ ℕ ∧ G ∈ ℕ)) → (((norm ‘(B −v A)) < (R + (1 / D)) ∧ (norm ‘(C −v A)) < (R + (1 / G))) → ((N ≤ D ∧ N ≤ G) → (norm ‘(B −v C)) < (√ ‘((4 · ((2 · R) + 1)) / N)))))
Theorem	projlem21 5213	Part of Lemma 3.6 of [Beran] p. 100. The hypothesis lets us work with our postulated vector sequence (whose existence was shown by projlem17 5209). Here we just show the sequence value belongs to the closed subspace H. Used by projlem27 5219 projlem28 5220.
⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))    ⇒   ⊢ (φ → (D ∈ ℕ → (F ‘D) ∈ H))
Theorem	projlem22 5214	Part of Lemma 3.6 of [Beran] p. 100. Here we show a member of the vector sequence is bounded. Used by projlem27 5219.
⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))    ⇒   ⊢ (φ → (D ∈ ℕ → (norm ‘((F ‘D) −v A)) < (R + (1 / D))))
Theorem	projlem23 5215	Part of Lemma 3.6 of [Beran] p. 101. The hypothesis lets us work with the sequence G which corresponds to Beran's "{||yn-x0||}". Used by projlem25 5217 projlem26 5218.
⊢ G = {⟨x, y⟩∣(x ∈ ℕ ∧ y = (norm ‘((F ‘x) −v A)))}    ⇒   ⊢ (D ∈ ℕ → (G ‘D) = (norm ‘((F ‘D) −v A)))
Theorem	projlem24 5216	Part of Lemma 3.6 of [Beran] p. 101. Here we show our vector sequence implies the real numbers sequence G corresponding to Beran's "{||yn-x0||}". Used by projlem25 5217 projlem26 5218.
⊢ G = {⟨x, y⟩∣(x ∈ ℕ ∧ y = (norm ‘((F ‘x) −v A)))}    &   ⊢ A ∈ ℋ    ⇒   ⊢ (F:ℕ–→ ℋ → G:ℕ–→ℂ)
Theorem	projlem25 5217	Part of Lemma 3.6 of [Beran] p. 101. "The sequence {||yn-x0||} of real numbers converges to the number ||y0-x0||" (here, "y0" is A and "x0" is z). Used by projlem31 5223.
⊢ G = {⟨x, y⟩∣(x ∈ ℕ ∧ y = (norm ‘((F ‘x) −v A)))}    &   ⊢ A ∈ ℋ    &   ⊢ F ∈ V    ⇒   ⊢ (F ⇝v z → G ⇝ (norm ‘(z −v A)))
Theorem	projlem26 5218	Part of Lemma 3.6 of [Beran] p. 101. The real sequence G (Beran's "{||yn-x0||}") converges to the infimum of norms. Used by projlem31 5223.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))    &   ⊢ G = {⟨x, y⟩∣(x ∈ ℕ ∧ y = (norm ‘((F ‘x) −v A)))}    ⇒   ⊢ (φ → G ⇝ R)
Theorem	projlem27 5219	Part of Lemma 3.6 of [Beran] p. 101. Boundedness to show F is a Cauchy sequence. Used by projlem28 5220.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))    &   ⊢ N ∈ ℕ    ⇒   ⊢ ((φ ∧ (D ∈ ℕ ∧ G ∈ ℕ)) → ((N ≤ D ∧ N ≤ G) → (norm ‘((F ‘D) −v (F ‘G))) < (√ ‘((4 · ((2 · R) + 1)) / N))))
Theorem	projlem28 5220	Part of Lemma 3.6 of [Beran] p. 101. Boundedness to show F is a Cauchy sequence. Used by projlem29 5221.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))    &   ⊢ D ∈ ℝ    ⇒   ⊢ (φ → (0 < D → ∃z ∈ ℕ ∀x ∈ ℕ ∀y ∈ ℕ ((z ≤ x ∧ z ≤ y) → (norm ‘((F ‘x) −v (F ‘y))) < D)))
Theorem	projlem29 5221	Part of Lemma 3.6 of [Beran] p. 101: 'Hence, {yn} is a Cauchy sequence.' Used by projlem30 5222.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))    ⇒   ⊢ (φ → F ∈ Cauchy)
Theorem	projlem30 5222	Part of Lemma 3.6 of [Beran] p. 101: 'By completeness of W, there exists y0 e. W such that yn->y0.' Used by projlem31 5223.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))    ⇒   ⊢ (φ → ∃z ∈ H F ⇝v z)
Theorem	projlem31 5223	Part of Lemma 3.6 of [Beran] p. 101. The postulated vector sequence F implies our conclusion. By showing such a sequence exists (which was done with the Axiom of Choice in projlem17 5209), we can show the final conclusion, projlem 5224.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ S = {u ∈ ℝ∣∃v ∈ H u = -(norm ‘(v −v A))}    &   ⊢ R = -sup(S, ℝ, < )    &   ⊢ (φ ↔ (F:ℕ–→H ∧ ∀w ∈ ℕ ((R − (1 / w)) < (norm ‘((F ‘w) −v A)) ∧ (norm ‘((F ‘w) −v A)) < (R + (1 / w)))))    &   ⊢ F ∈ V    ⇒   ⊢ (φ → ∃x ∈ H ∀y ∈ H (norm ‘(x −v A)) ≤ (norm ‘(y −v A)))
Theorem	projlem 5224	Lemma 3.6 of [Beran] p. 101: "Let H be a complete subspace of a (pre-)Hilbert space ℋ and let A ∈ ℋ. Then there exists a vector x ∈ H such that (norm ‘(x −v A)) ≤ (norm ‘(y −v A)) for every y ∈ H." This is a lemma for the projection theorem.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    ⇒   ⊢ ∃x ∈ H ∀y ∈ H (norm ‘(x −v A)) ≤ (norm ‘(y −v A))
Theorem	pjthlem1 5225	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ A ∈ ℋ    &   ⊢ B ∈ ℋ    &   ⊢ D ∈ ℋ    &   ⊢ S ∈ ℂ    &   ⊢ C = (A −v B)    ⇒   ⊢ ((norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A)) ↔ ((norm ‘C)↑2) ≤ ((norm ‘(C −v (S ·s D)))↑2))
Theorem	pjthlem2 5226	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    ⇒   ⊢ (¬ D = 0v → R ∈ ℝ)
Theorem	pjthlem3 5227	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    ⇒   ⊢ (¬ D = 0v → 0 < R)
Theorem	pjthlem4 5228	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    &   ⊢ C ∈ ℋ    &   ⊢ S = (R · (C ·i D))    ⇒   ⊢ (¬ D = 0v → S ∈ ℂ)
Theorem	pjthlem5 5229	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ D ∈ ℋ    &   ⊢ C ∈ ℋ    &   ⊢ S ∈ ℂ    ⇒   ⊢ ((norm ‘(C −v (S ·s D)))↑2) = ((((norm ‘C)↑2) − (∗ ‘(S · (D ·i C)))) + (((S · (∗ ‘S)) · (D ·i D)) − (S · (D ·i C))))
Theorem	pjthlem6 5230	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    &   ⊢ C ∈ ℋ    &   ⊢ S = (R · (C ·i D))    ⇒   ⊢ (¬ D = 0v → (S · (D ·i C)) = (R · ((abs ‘(C ·i D))↑2)))
Theorem	pjthlem7 5231	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    &   ⊢ C ∈ ℋ    &   ⊢ S = (R · (C ·i D))    ⇒   ⊢ (¬ D = 0v → ((S · (∗ ‘S)) · (D ·i D)) = (R · ((abs ‘(C ·i D))↑2)))
Theorem	pjthlem8 5232	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    &   ⊢ C ∈ ℋ    &   ⊢ S = (R · (C ·i D))    ⇒   ⊢ (¬ D = 0v → ((norm ‘(C −v (S ·s D)))↑2) = (((norm ‘C)↑2) − (R · ((abs ‘(C ·i D))↑2))))
Theorem	pjthlem9 5233	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ A ∈ ℋ    &   ⊢ B ∈ ℋ    &   ⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    &   ⊢ S = (R · (C ·i D))    &   ⊢ C = (A −v B)    ⇒   ⊢ (¬ D = 0v → ((norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A)) ↔ ((norm ‘C)↑2) ≤ ((norm ‘(C −v (S ·s D)))↑2)))
Theorem	pjthlem10 5234	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ A ∈ ℋ    &   ⊢ B ∈ ℋ    &   ⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    &   ⊢ S = (R · (C ·i D))    &   ⊢ C = (A −v B)    ⇒   ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → (R · ((abs ‘(C ·i D))↑2)) = 0)
Theorem	pjthlem11 5235	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ A ∈ ℋ    &   ⊢ B ∈ ℋ    &   ⊢ D ∈ ℋ    &   ⊢ R = (1 / (D ·i D))    &   ⊢ S = (R · (C ·i D))    &   ⊢ C = (A −v B)    ⇒   ⊢ ((¬ D = 0v ∧ (norm ‘(B −v A)) ≤ (norm ‘((B +v (S ·s D)) −v A))) → (C ·i D) = 0)
Theorem	pjthlem12 5236	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ B ∈ H    &   ⊢ D ∈ H    &   ⊢ R = (1 / (D ·i D))    &   ⊢ S = (R · (C ·i D))    &   ⊢ C = (A −v B)    ⇒   ⊢ (∀y ∈ H (norm ‘(B −v A)) ≤ (norm ‘(y −v A)) → (¬ D = 0v → (C ·i D) = 0))
Theorem	pjthlem13 5237	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ B ∈ H    &   ⊢ D ∈ H    &   ⊢ C = (A −v B)    ⇒   ⊢ (∀y ∈ H (norm ‘(B −v A)) ≤ (norm ‘(y −v A)) → (C ·i D) = 0)
Theorem	pjthlem14 5238	Lemma for Theorem 3.7(i) of [Beran] p. 102.
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    &   ⊢ B ∈ H    &   ⊢ C = (A −v B)    ⇒   ⊢ (∀z ∈ H (norm ‘(B −v A)) ≤ (norm ‘(z −v A)) → ∃x ∈ H ∃y ∈ (⊥ ‘H)A = (x +v y))
Theorem	pjth 5239	Projection Theorem: Any Hilbert space vector A can be decomposed into a member x of a closed subspace H and a member y of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part).
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    ⇒   ⊢ ∃x ∈ H ∃y ∈ (⊥ ‘H)A = (x +v y)
Theorem	pjtht 5240	Projection Theorem: Any Hilbert space vector A can be decomposed into a member x of a closed subspace H and a member y of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part).
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → ∃x ∈ H ∃y ∈ (⊥ ‘H)A = (x +v y))
Theorem	pjthu 5241	Corollary of projection theorem.
⊢ H ∈ Cℋ    ⇒   ⊢ (A ∈ ℋ → ∃!x ∈ H ∃y ∈ (⊥ ‘H)A = (x +v y))
Theorem	pjthu2 5242	Corollary of projection theorem.
⊢ H ∈ Cℋ    ⇒   ⊢ (A ∈ ℋ → ∃!y ∈ (⊥ ‘H)∃x ∈ H A = (x +v y))
Theorem	pjthut 5243	Corollary of projection theorem.
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → ∃!x ∈ H ∃y ∈ (⊥ ‘H)A = (x +v y))
Definition	df-pj 5244	Define the projection map on a Hilbert space, as a mapping from the Hilbert lattice to a function on Hilbert space. Every closed subspace is associated with a unique projection. Remark of [Kalmbach] p. 66, adopted as a definition.
⊢ Proj = {⟨h, f⟩∣(h ∈ Cℋ ∧ f = {⟨x, y⟩∣(x ∈ ℋ ∧ y = ∪{z ∈ h∣∃w ∈ (⊥ ‘h)x = (z +v w)})})}
Theorem	pjmvalt 5245	The value of the projection map.
⊢ (H ∈ Cℋ → (Proj ‘H) = {⟨x, y⟩∣(x ∈ ℋ ∧ y = ∪{z ∈ H∣∃w ∈ (⊥ ‘H)x = (z +v w)})})
Theorem	pjvalt 5246	Value of a projection.
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → ((Proj ‘H) ‘A) = ∪{x ∈ H∣∃y ∈ (⊥ ‘H)A = (x +v y)})
Theorem	axpjclt 5247	Closure of a projection in its subspace. If we consider this together with axpjpjt 5260 to be axioms, the need for the ax-hcompl 5113 can often be avoided for the kinds of theorems we are interested in here. An interesting project is to see how far we can go by using them in place of it. In particular, we can prove the orthomodular law pjoml 5271.)
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → ((Proj ‘H) ‘A) ∈ H)
Theorem	pjhclt 5248	Closure of a projection in Hilbert space.
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → ((Proj ‘H) ‘A) ∈ ℋ )
Theorem	omlsilem 5249	Lemma for orthomodular law in the Hilbert lattice.
⊢ G ∈ Sℋ    &   ⊢ H ∈ Sℋ    &   ⊢ G ⊆ H    &   ⊢ (H ∩ (⊥ ‘G)) = 0ℋ    &   ⊢ A ∈ H    &   ⊢ B ∈ G    &   ⊢ C ∈ (⊥ ‘G)    ⇒   ⊢ (A = (B +v C) → A ∈ G)
Theorem	omlsi 5250	Subspace inference form of orthomodular law in the Hilbert lattice.
⊢ A ∈ Cℋ    &   ⊢ B ∈ Sℋ    &   ⊢ A ⊆ B    &   ⊢ (B ∩ (⊥ ‘A)) = 0ℋ    ⇒   ⊢ A = B
Theorem	omls 5251	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22.
⊢ A ∈ Cℋ    &   ⊢ B ∈ Sℋ    ⇒   ⊢ ((A ⊆ B ∧ (B ∩ (⊥ ‘A)) = 0ℋ) → A = B)
Theorem	ococ 5252	Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102.
⊢ A ∈ Cℋ    ⇒   ⊢ (⊥ ‘(⊥ ‘A)) = A
Theorem	ococt 5253	Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102.
⊢ (A ∈ Cℋ → (⊥ ‘(⊥ ‘A)) = A)
Theorem	dfch2 5254	Alternate definition of the Hilbert lattice.
⊢ Cℋ = {x∣(x ⊆ ℋ ∧ (⊥ ‘(⊥ ‘x)) = x)}
Theorem	pjcl 5255	Closure of a projection in its subspace.
⊢ H ∈ Cℋ    ⇒   ⊢ (A ∈ ℋ → ((Proj ‘H) ‘A) ∈ H)
Theorem	pjhcl 5256	Closure of a projection in Hilbert space.
⊢ H ∈ Cℋ    ⇒   ⊢ (A ∈ ℋ → ((Proj ‘H) ‘A) ∈ ℋ )
Theorem	pjcli 5257	Closure of a projection in its subspace.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ ((Proj ‘H) ‘A) ∈ H
Theorem	pjhcli 5258	Closure of a projection in Hilbert space.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ ((Proj ‘H) ‘A) ∈ ℋ
Theorem	pjpj0 5259	Decomposition of a vector into projections.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ A = (((Proj ‘H) ‘A) +v ((Proj ‘(⊥ ‘H)) ‘A))
Theorem	axpjpjt 5260	Decomposition of a vector into projections. See comment in axpjclt 5247.
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → A = (((Proj ‘H) ‘A) +v ((Proj ‘(⊥ ‘H)) ‘A)))
Theorem	pjpj 5261	Decomposition of a vector into projections.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ A = (((Proj ‘H) ‘A) +v ((Proj ‘(⊥ ‘H)) ‘A))
Theorem	pjpjtht 5262	Projection Theorem: Any Hilbert space vector A can be decomposed into a member x of a closed subspace H and a member y of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part).
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → ∃x ∈ H ∃y ∈ (⊥ ‘H)A = (x +v y))
Theorem	pjpjth 5263	Projection Theorem: Any Hilbert space vector A can be decomposed into a member x of a closed subspace H and a member y of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part).
⊢ A ∈ ℋ    &   ⊢ H ∈ Cℋ    ⇒   ⊢ ∃x ∈ H ∃y ∈ (⊥ ‘H)A = (x +v y)
Theorem	pjopt 5264	Orthocomplement projection in terms of projection.
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → ((Proj ‘(⊥ ‘H)) ‘A) = (A −v ((Proj ‘H) ‘A)))
Theorem	pjpot 5265	Projection in terms of orthocomplement projection.
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → ((Proj ‘H) ‘A) = (A −v ((Proj ‘(⊥ ‘H)) ‘A)))
Theorem	pjop 5266	Orthocomplement projection in terms of projection.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ ((Proj ‘(⊥ ‘H)) ‘A) = (A −v ((Proj ‘H) ‘A))
Theorem	pjpo 5267	Projection in terms of orthocomplement projection.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ ((Proj ‘H) ‘A) = (A −v ((Proj ‘(⊥ ‘H)) ‘A))
Theorem	pjoc1 5268	Projection of a vector in the orthocomplement of the projection subspace.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ (A ∈ H ↔ ((Proj ‘(⊥ ‘H)) ‘A) = 0v)
Theorem	pjch 5269	Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ (A ∈ H ↔ ((Proj ‘H) ‘A) = A)
Theorem	pjoc1t 5270	Projection of a vector in the orthocomplement of the projection subspace.
⊢ ((H ∈ Cℋ ∧ A ∈ ℋ ) → (A ∈ H ↔ ((Proj ‘(⊥ ‘H)) ‘A) = 0v))
Theorem	pjoml 5271	Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. Derived using projections; compare omls 5251.
⊢ A ∈ Cℋ    &   ⊢ B ∈ Sℋ    ⇒   ⊢ ((A ⊆ B ∧ (B ∩ (⊥ ‘A)) = 0ℋ) → A = B)
Theorem	pjococ 5272	Proof of orthocomplement theorem using projections. Compare ococ 5252.
⊢ H ∈ Cℋ    ⇒   ⊢ (⊥ ‘(⊥ ‘H)) = H
Theorem	pjoc2 5273	Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111.
⊢ H ∈ Cℋ    &   ⊢ A ∈ ℋ    ⇒   ⊢ (A ∈ (⊥ ‘H) ↔ ((Proj ‘H) ‘A) = 0v)
Theorem	pjoc2t 5274	Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111.
⊢ H ∈ Cℋ    ⇒   ⊢ (A ∈ ℋ → (A ∈ (⊥ ‘H) ↔ ((Proj ‘H) ‘A) = 0v))
Definition	df-shsum 5275	Define subspace sum in Sℋ. See shsumvalt 5279, shsumval2 5361, and shsumval3 5362 for its value.
⊢ +ℋ = {⟨⟨x, y⟩, z⟩∣((x ∈ Sℋ ∧ y ∈ Sℋ ) ∧ z = {w ∈ ℋ ∣∃v ∈ x ∃u ∈ y w = (v +v u)})}
Definition	df-span 5276	Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanvalt 5300 for its value.
⊢ span = {⟨x, y⟩∣(x ⊆ ℋ ∧ y = ∩{z ∈ Sℋ ∣x ⊆ z})}
Definition	df-chj 5277	Define Hilbert lattice join. See chjvalt 5323 for its value and chjclt 5330 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to Cℋ; see sshjclt 5328. For an alternate definition see dfchj2 5325.
⊢ ∨ℋ = {⟨⟨x