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Theorem sshjval 31286
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 30935 . . 3 ℋ ∈ V
21elpw2 5354 . 2 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 5354 . 2 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uneq12 4158 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
54fveq2d 6907 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (⊥‘(𝑥𝑦)) = (⊥‘(𝐴𝐵)))
65fveq2d 6907 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (⊥‘(⊥‘(𝑥𝑦))) = (⊥‘(⊥‘(𝐴𝐵))))
7 df-chj 31246 . . 3 = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥𝑦))))
8 fvex 6916 . . 3 (⊥‘(⊥‘(𝐴𝐵))) ∈ V
96, 7, 8ovmpoa 7583 . 2 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
102, 3, 9syl2anbr 597 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cun 3945  wss 3947  𝒫 cpw 4607  cfv 6556  (class class class)co 7426  chba 30855  cort 30866   chj 30869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-hilex 30935
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6508  df-fun 6558  df-fv 6564  df-ov 7429  df-oprab 7430  df-mpo 7431  df-chj 31246
This theorem is referenced by:  shjval  31287  sshjval3  31290  sshjcl  31291  sshjval2  31347  ssjo  31383  sshhococi  31482
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