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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlatexchb1 | Structured version Visualization version GIF version |
Description: A version of cvlexchb1 36470 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 36465 | . . . . 5 ⊢ (𝐾 ∈ CvLat → 𝐾 ∈ AtLat) | |
2 | 1 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ AtLat) |
3 | simpr1 1190 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑃 ∈ 𝐴) | |
4 | simpr3 1192 | . . . 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 36452 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑅 ↔ 𝑃 ≠ 𝑅)) |
8 | 2, 3, 4, 7 | syl3anc 1367 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (¬ 𝑃 ≤ 𝑅 ↔ 𝑃 ≠ 𝑅)) |
9 | eqid 2824 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | 9, 6 | atbase 36429 | . . . 4 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾)) |
11 | cvlatexch.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
12 | 9, 5, 11, 6 | cvlexchb1 36470 | . . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ (Base‘𝐾)) ∧ ¬ 𝑃 ≤ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄))) |
13 | 12 | 3expia 1117 | . . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ (Base‘𝐾))) → (¬ 𝑃 ≤ 𝑅 → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))) |
14 | 10, 13 | syl3anr3 1414 | . . 3 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (¬ 𝑃 ≤ 𝑅 → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))) |
15 | 8, 14 | sylbird 262 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃 ≠ 𝑅 → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄)))) |
16 | 15 | 3impia 1113 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 lecple 16575 joincjn 17557 Atomscatm 36403 AtLatcal 36404 CvLatclc 36405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-lat 17659 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 |
This theorem is referenced by: cvlatexchb2 36475 cvlatexch1 36476 cvlatexch3 36478 hlatexchb1 36533 llnexchb2lem 37008 4atexlemunv 37206 cdleme19d 37446 |
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