Metamath Proof Explorer

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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.


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.


Theorem	projlem11 5203	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). 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.


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.


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.


Theorem	projlem14 5206	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 5208.


Theorem	projlem15 5207	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 5208.


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.
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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 {y_n } be a sequence of W such that i₀ - 1/n < \|\|x₀ - y_n \|\| < i₀ + 1/n." Here, corresponds to "W"; to "{y_n }"; to "n"; to "i₀ "; and to "\|\|x₀ - y_n \|\|". Used by projlem 5224.
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Theorem	projlem18 5210	Part of Lemma 3.6 of [Beran] p. 101, top. Used by projlem19 5211.


Theorem	projlem19 5211	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem20 5212.
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Theorem	projlem20 5212	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem27 5219.
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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 . Used by projlem27 5219 projlem28 5220.
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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.
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Theorem	projlem23 5215	Part of Lemma 3.6 of [Beran] p. 101. The hypothesis lets us work with the sequence which corresponds to Beran's "{\|\|yn-x0\|\|}". Used by projlem25 5217 projlem26 5218.
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Theorem	projlem24 5216	Part of Lemma 3.6 of [Beran] p. 101. Here we show our vector sequence implies the real numbers sequence corresponding to Beran's "{\|\|yn-x0\|\|}". Used by projlem25 5217 projlem26 5218.
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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 and "x0" is ). Used by projlem31 5223.
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Theorem	projlem26 5218	Part of Lemma 3.6 of [Beran] p. 101. The real sequence (Beran's "{\|\|yn-x0\|\|}") converges to the infimum of norms. Used by projlem31 5223.
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Theorem	projlem27 5219	Part of Lemma 3.6 of [Beran] p. 101. Boundedness to show is a Cauchy sequence. Used by projlem28 5220.
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