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Theorem sshjval 31384
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 31033 . . 3 ℋ ∈ V
21elpw2 5352 . 2 (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
31elpw2 5352 . 2 (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4 uneq12 4186 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
54fveq2d 6926 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (⊥‘(𝑥𝑦)) = (⊥‘(𝐴𝐵)))
65fveq2d 6926 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (⊥‘(⊥‘(𝑥𝑦))) = (⊥‘(⊥‘(𝐴𝐵))))
7 df-chj 31344 . . 3 = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥𝑦))))
8 fvex 6935 . . 3 (⊥‘(⊥‘(𝐴𝐵))) ∈ V
96, 7, 8ovmpoa 7607 . 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 1537  wcel 2108  cun 3974  wss 3976  𝒫 cpw 4622  cfv 6575  (class class class)co 7450  chba 30953  cort 30964   chj 30967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-hilex 31033
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-chj 31344
This theorem is referenced by:  shjval  31385  sshjval3  31388  sshjcl  31389  sshjval2  31445  ssjo  31481  sshhococi  31580
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