Theorem sshjval 28908

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 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))

Proof of Theorem sshjval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 28555	. . 3 ⊢ ℋ ∈ V
2	1	elpw2 5104	. 2 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
3	1	elpw2 5104	. 2 ⊢ (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ)
4		uneq12 4024	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝐵))
5	4	fveq2d 6503	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⊥‘(𝑥 ∪ 𝑦)) = (⊥‘(𝐴 ∪ 𝐵)))
6	5	fveq2d 6503	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⊥‘(⊥‘(𝑥 ∪ 𝑦))) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
7		df-chj 28868	. . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦))))
8		fvex 6512	. . 3 ⊢ (⊥‘(⊥‘(𝐴 ∪ 𝐵))) ∈ V
9	6, 7, 8	ovmpoa 7121	. 2 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))
10	2, 3, 9	syl2anbr 589	1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵))))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ∪ cun 3828   ⊆ wss 3830  𝒫 cpw 4422  ‘cfv 6188  (class class class)co 6976   ℋchba 28475  ⊥cort 28486   ∨_ℋ chj 28489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-hilex 28555

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-chj 28868

This theorem is referenced by:  shjval  28909  sshjval3  28912  sshjcl  28913  sshjval2  28969  ssjo  29005  sshhococi  29104