![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > shjval | Structured version Visualization version GIF version |
Description: Value of join in Sℋ. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shjval | ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shss 30887 | . 2 ⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ ℋ) | |
2 | shss 30887 | . 2 ⊢ (𝐵 ∈ Sℋ → 𝐵 ⊆ ℋ) | |
3 | sshjval 31027 | . 2 ⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) | |
4 | 1, 2, 3 | syl2an 595 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (⊥‘(⊥‘(𝐴 ∪ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∪ cun 3938 ⊆ wss 3940 ‘cfv 6533 (class class class)co 7401 ℋchba 30596 Sℋ csh 30605 ⊥cort 30607 ∨ℋ chj 30610 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-hilex 30676 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-sh 30884 df-chj 30987 |
This theorem is referenced by: chjval 31029 shjcom 31035 shlej1 31037 shunssji 31046 shlub 31091 shjshsi 31169 |
Copyright terms: Public domain | W3C validator |