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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 31022 | . . 3 ⊢ ℋ ∈ V | |
2 | 1 | elpw2 5355 | . 2 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ) |
3 | 1 | elpw2 5355 | . 2 ⊢ (𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ) |
4 | uneq12 4180 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∪ 𝑦) = (𝐴 ∪ 𝐵)) | |
5 | 4 | fveq2d 6923 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⊥‘(𝑥 ∪ 𝑦)) = (⊥‘(𝐴 ∪ 𝐵))) |
6 | 5 | fveq2d 6923 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (⊥‘(⊥‘(𝑥 ∪ 𝑦))) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
7 | df-chj 31333 | . . 3 ⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦)))) | |
8 | fvex 6932 | . . 3 ⊢ (⊥‘(⊥‘(𝐴 ∪ 𝐵))) ∈ V | |
9 | 6, 7, 8 | ovmpoa 7601 | . 2 ⊢ ((𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
10 | 2, 3, 9 | syl2anbr 598 | 1 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ∪ cun 3968 ⊆ wss 3970 𝒫 cpw 4622 ‘cfv 6572 (class class class)co 7445 ℋchba 30942 ⊥cort 30953 ∨ℋ chj 30956 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-hilex 31022 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-iota 6524 df-fun 6574 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-chj 31333 |
This theorem is referenced by: shjval 31374 sshjval3 31377 sshjcl 31378 sshjval2 31434 ssjo 31470 sshhococi 31569 |
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