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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatexch2 | Structured version Visualization version GIF version |
Description: Atom exchange property. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlatexch.l | ⊢ ≤ = (le‘𝐾) |
cvlatexch.j | ⊢ ∨ = (join‘𝐾) |
cvlatexch.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvlatexch2 | ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlatexch.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | cvlatexch.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | cvlatexch.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 1, 2, 3 | cvlatexch1 35144 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) → 𝑄 ≤ (𝑅 ∨ 𝑃))) |
5 | cvllat 35134 | . . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ Lat) | |
6 | 5 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → 𝐾 ∈ Lat) |
7 | simp22 1250 | . . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → 𝑄 ∈ 𝐴) | |
8 | eqid 2760 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
9 | 8, 3 | atbase 35097 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → 𝑄 ∈ (Base‘𝐾)) |
11 | simp23 1251 | . . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → 𝑅 ∈ 𝐴) | |
12 | 8, 3 | atbase 35097 | . . . . 5 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
13 | 11, 12 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → 𝑅 ∈ (Base‘𝐾)) |
14 | 8, 2 | latjcom 17280 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄)) |
15 | 6, 10, 13, 14 | syl3anc 1477 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄)) |
16 | 15 | breq2d 4816 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ 𝑃 ≤ (𝑅 ∨ 𝑄))) |
17 | simp21 1249 | . . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → 𝑃 ∈ 𝐴) | |
18 | 8, 3 | atbase 35097 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
19 | 17, 18 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → 𝑃 ∈ (Base‘𝐾)) |
20 | 8, 2 | latjcom 17280 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃)) |
21 | 6, 19, 13, 20 | syl3anc 1477 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃)) |
22 | 21 | breq2d 4816 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑄 ≤ (𝑃 ∨ 𝑅) ↔ 𝑄 ≤ (𝑅 ∨ 𝑃))) |
23 | 4, 16, 22 | 3imtr4d 283 | 1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 Basecbs 16079 lecple 16170 joincjn 17165 Latclat 17266 Atomscatm 35071 CvLatclc 35073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-preset 17149 df-poset 17167 df-plt 17179 df-lub 17195 df-glb 17196 df-join 17197 df-meet 17198 df-p0 17260 df-lat 17267 df-covers 35074 df-ats 35075 df-atl 35106 df-cvlat 35130 |
This theorem is referenced by: hlatexch2 35203 4atexlemnclw 35877 4atexlemex2 35878 cdleme21ct 36137 cdleme22f 36154 cdleme22f2 36155 cdlemf1 36369 |
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