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Mirrors > Home > HSE Home > Th. List > sshjval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
sshjval | ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex 30829 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | elpw2 5351 | . 2 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ) |
3 | 1 | elpw2 5351 | . 2 ⊢ (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ) |
4 | uneq12 4159 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
5 | 4 | fveq2d 6906 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⊥‘(𝑥 ∪ 𝑦)) = (⊥‘(𝐴 ∪ 𝐵))) |
6 | 5 | fveq2d 6906 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⊥‘(⊥‘(𝑥 ∪ 𝑦))) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
7 | df-chj 31140 | . . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦)))) | |
8 | fvex 6915 | . . 3 ⊢ (⊥‘(⊥‘(𝐴 ∪ 𝐵))) ∈ V | |
9 | 6, 7, 8 | ovmpoa 7582 | . 2 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
10 | 2, 3, 9 | syl2anbr 597 | 1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3947 ⊆ wss 3949 𝒫 cpw 4606 ‘cfv 6553 (class class class)co 7426 ℋchba 30749 ⊥cort 30760 ∨ℋ chj 30763 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-hilex 30829 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-chj 31140 |
This theorem is referenced by: shjval 31181 sshjval3 31184 sshjcl 31185 sshjval2 31241 ssjo 31277 sshhococi 31376 |
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