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Mirrors > Home > HSE Home > Th. List > chjcom | Structured version Visualization version GIF version |
Description: Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chjcom | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsh 29007 | . 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
2 | chsh 29007 | . 2 ⊢ (𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ ) | |
3 | shjcom 29141 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)) | |
4 | 1, 2, 3 | syl2an 598 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 Sℋ csh 28711 Cℋ cch 28712 ∨ℋ chj 28716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-hilex 28782 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-sh 28990 df-ch 29004 df-chj 29093 |
This theorem is referenced by: chub2 29291 chlejb2 29296 chj12 29317 mddmd2 30092 dmdsl3 30098 csmdsymi 30117 mdexchi 30118 atordi 30167 atcvatlem 30168 atcvati 30169 chirredlem2 30174 chirredlem4 30176 atcvat3i 30179 atcvat4i 30180 atdmd 30181 mdsymlem3 30188 mdsymlem5 30190 mdsymlem8 30193 sumdmdlem2 30202 dmdbr5ati 30205 |
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