Theorem shsumvalt 9272

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 +h z)})

Distinct variable groups:   x,y,z,A   x,B,y,z

Proof of Theorem shsumvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8864	. . 3 ⊢ ℋ ∈ V
2	1	rabex 2730	. 2 ⊢ {x ∈ ℋ ∣∃y ∈ A ∃z ∈ B x = (y +h z)} ∈ V
3		rexeq1 1790	. . 3 ⊢ (w = A → (∃y ∈ w ∃z ∈ v x = (y +h z) ↔ ∃y ∈ A ∃z ∈ v x = (y +h z)))
4	3	rabbisdv 1810	. 2 ⊢ (w = A → {x ∈ ℋ ∣∃y ∈ w ∃z ∈ v x = (y +h z)} = {x ∈ ℋ ∣∃y ∈ A ∃z ∈ v x = (y +h z)})
5		rexeq1 1790	. . . 4 ⊢ (v = B → (∃z ∈ v x = (y +h z) ↔ ∃z ∈ B x = (y +h z)))
6	5	rexbidv 1667	. . 3 ⊢ (v = B → (∃y ∈ A ∃z ∈ v x = (y +h z) ↔ ∃y ∈ A ∃z ∈ B x = (y +h z)))
7	6	rabbisdv 1810	. 2 ⊢ (v = B → {x ∈ ℋ ∣∃y ∈ A ∃z ∈ v x = (y +h z)} = {x ∈ ℋ ∣∃y ∈ A ∃z ∈ B x = (y +h z)})
8		df-shsum 9268	. 2 ⊢ +ℋ = {⟨⟨w, v⟩, u⟩∣((w ∈ Sℋ ⋀ v ∈ Sℋ ) ⋀ u = {x ∈ ℋ ∣∃y ∈ w ∃z ∈ v x = (y +h z)})}
9	2, 4, 7, 8	oprabval2 4034	1 ⊢ ((A ∈ Sℋ ⋀ B ∈ Sℋ ) → (A +ℋ B) = {x ∈ ℋ ∣∃y ∈ A ∃z ∈ B x = (y +h z)})

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃wrex 1649  {crab 1651  (class class class)co 3969   ℋ chil 8783   +_h cva 8784   S_ℋ csh 8792   +_ℋ cph 8795

This theorem is referenced by:  shselt 9273  shscl 9276

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opr 3971  df-oprab 3972  df-shsum 9268

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