Theorem shsumvalt 5279

Description: Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65.

Assertion

Ref	Expression
shsumvalt	⊢ ((A ∈ Sℋ ∧ B ∈ Sℋ ) → (A +ℋ B) = {x ∈ ℋ ∣∃y ∈ A ∃z ∈ B x = (y +v z)})

Distinct variable group(s):   x,y,z,A   x,B,y,z

Proof of Theorem shsumvalt

Step	Hyp	Ref	Expression
1		ax-hilex 4983	. . 3 ⊢ ℋ ∈ V
2	1	rabex 1706	. 2 ⊢ {x ∈ ℋ ∣∃y ∈ A ∃z ∈ B x = (y +v z)} ∈ V
3		rexeq 1325	. . 3 ⊢ (w = A → (∃y ∈ w ∃z ∈ v x = (y +v z) ↔ ∃y ∈ A ∃z ∈ v x = (y +v z)))
4	3	birabsdv 1344	. 2 ⊢ (w = A → {x ∈ ℋ ∣∃y ∈ w ∃z ∈ v x = (y +v z)} = {x ∈ ℋ ∣∃y ∈ A ∃z ∈ v x = (y +v z)})
5		rexeq 1325	. . . 4 ⊢ (v = B → (∃z ∈ v x = (y +v z) ↔ ∃z ∈ B x = (y +v z)))
6	5	birexdv 1220	. . 3 ⊢ (v = B → (∃y ∈ A ∃z ∈ v x = (y +v z) ↔ ∃y ∈ A ∃z ∈ B x = (y +v z)))
7	6	birabsdv 1344	. 2 ⊢ (v = B → {x ∈ ℋ ∣∃y ∈ A ∃z ∈ v x = (y +v z)} = {x ∈ ℋ ∣∃y ∈ A ∃z ∈ B x = (y +v z)})
8		df-shsum 5275	. 2 ⊢ +ℋ = {⟨⟨w, v⟩, u⟩∣((w ∈ Sℋ ∧ v ∈ Sℋ ) ∧ u = {x ∈ ℋ ∣∃y ∈ w ∃z ∈ v x = (y +v z)})}
9	2, 4, 7, 8	oprabval2 3051	1 ⊢ ((A ∈ Sℋ ∧ B ∈ Sℋ ) → (A +ℋ B) = {x ∈ ℋ ∣∃y ∈ A ∃z ∈ B x = (y +v z)})

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  (class class class)co 3001   ℋ chil 4958   +_v cva 4959   S_ℋ csh 4967   +_ℋ cph 4970

This theorem is referenced by:  shselt 5280  shscl 5282

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004  df-shsum 5275

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