Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatexchb1 | Structured version Visualization version GIF version |
Description: A version of cvlexchb1 36967 for atoms. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlatexch.l | ⊢ ≤ = (le‘𝐾) |
cvlatexch.j | ⊢ ∨ = (join‘𝐾) |
cvlatexch.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
cvlatexchb1 | ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvlatl 36962 | . . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ AtLat) |
3 | simpr1 1195 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑃 ∈ 𝐴) | |
4 | simpr3 1197 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ 𝐴) | |
5 | cvlatexch.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
6 | cvlatexch.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 5, 6 | atncmp 36949 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑅 ↔ 𝑃 ≠ 𝑅)) |
8 | 2, 3, 4, 7 | syl3anc 1372 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (¬ 𝑃 ≤ 𝑅 ↔ 𝑃 ≠ 𝑅)) |
9 | eqid 2738 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | 9, 6 | atbase 36926 | . . . 4 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
11 | cvlatexch.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
12 | 9, 5, 11, 6 | cvlexchb1 36967 | . . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ (Base‘𝐾)) ∧ ¬ 𝑃 ≤ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄))) |
13 | 12 | 3expia 1122 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ (Base‘𝐾))) → (¬ 𝑃 ≤ 𝑅 → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))) |
14 | 10, 13 | syl3anr3 1419 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (¬ 𝑃 ≤ 𝑅 → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))) |
15 | 8, 14 | sylbird 263 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃 ≠ 𝑅 → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))) |
16 | 15 | 3impia 1118 | 1 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 lecple 16675 joincjn 17670 Atomscatm 36900 AtLatcal 36901 CvLatclc 36902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-lat 17772 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 |
This theorem is referenced by: cvlatexchb2 36972 cvlatexch1 36973 cvlatexch3 36975 hlatexchb1 37030 llnexchb2lem 37505 4atexlemunv 37703 cdleme19d 37943 |
Copyright terms: Public domain | W3C validator |