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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 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)))) |
Colors of variables: wff setvar class |
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 |
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