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Theorem sshjval 28514
 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 28161 . . 3 ℋ ∈ V
21elpw2 4973 . 2 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 4973 . 2 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uneq12 3901 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
54fveq2d 6352 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (⊥‘(𝑥𝑦)) = (⊥‘(𝐴𝐵)))
65fveq2d 6352 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (⊥‘(⊥‘(𝑥𝑦))) = (⊥‘(⊥‘(𝐴𝐵))))
7 df-chj 28474 . . 3 = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥𝑦))))
8 fvex 6358 . . 3 (⊥‘(⊥‘(𝐴𝐵))) ∈ V
96, 7, 8ovmpt2a 6952 . 2 ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
102, 3, 9syl2anbr 498 1 ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1628   ∈ wcel 2135   ∪ cun 3709   ⊆ wss 3711  𝒫 cpw 4298  ‘cfv 6045  (class class class)co 6809   ℋchil 28081  ⊥cort 28092   ∨ℋ chj 28095 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051  ax-hilex 28161 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-iota 6008  df-fun 6047  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-chj 28474 This theorem is referenced by:  shjval  28515  sshjval3  28518  sshjcl  28519  sshjval2  28575  ssjo  28611  sshhococi  28710
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