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Theorem sshjval 31107
Description: Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
sshjval ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))

Proof of Theorem sshjval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hilex 30756 . . 3 ℋ ∈ V
21elpw2 5338 . 2 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 5338 . 2 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uneq12 4153 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
54fveq2d 6888 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (⊥‘(𝑥𝑦)) = (⊥‘(𝐴𝐵)))
65fveq2d 6888 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (⊥‘(⊥‘(𝑥𝑦))) = (⊥‘(⊥‘(𝐴𝐵))))
7 df-chj 31067 . . 3 = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥𝑦))))
8 fvex 6897 . . 3 (⊥‘(⊥‘(𝐴𝐵))) ∈ V
96, 7, 8ovmpoa 7558 . 2 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
102, 3, 9syl2anbr 598 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cun 3941  wss 3943  𝒫 cpw 4597  cfv 6536  (class class class)co 7404  chba 30676  cort 30687   chj 30690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-hilex 30756
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-chj 31067
This theorem is referenced by:  shjval  31108  sshjval3  31111  sshjcl  31112  sshjval2  31168  ssjo  31204  sshhococi  31303
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