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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme21g | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.) |
Ref | Expression |
---|---|
cdleme21.l | ⊢ ≤ = (le‘𝐾) |
cdleme21.j | ⊢ ∨ = (join‘𝐾) |
cdleme21.m | ⊢ ∧ = (meet‘𝐾) |
cdleme21.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme21.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme21.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme21.f | ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) |
cdleme21g.g | ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) |
cdleme21g.d | ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) |
cdleme21g.y | ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) |
cdleme21g.n | ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) |
cdleme21g.o | ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) |
Ref | Expression |
---|---|
cdleme21g | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑁 = 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme21.l | . 2 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme21.j | . 2 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme21.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleme21.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdleme21.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdleme21.u | . 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
7 | cdleme21.f | . 2 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) | |
8 | eqid 2818 | . 2 ⊢ ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) | |
9 | cdleme21g.d | . 2 ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) | |
10 | eqid 2818 | . 2 ⊢ ((𝑅 ∨ 𝑧) ∧ 𝑊) = ((𝑅 ∨ 𝑧) ∧ 𝑊) | |
11 | cdleme21g.n | . 2 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) | |
12 | eqid 2818 | . 2 ⊢ ((𝑃 ∨ 𝑄) ∧ (((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑧) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑧) ∧ 𝑊))) | |
13 | cdleme21g.g | . 2 ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) | |
14 | cdleme21g.y | . 2 ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) | |
15 | cdleme21g.o | . 2 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | cdleme21f 37348 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑈 ≤ (𝑆 ∨ 𝑇)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))) → 𝑁 = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 lecple 16560 joincjn 17542 meetcmee 17543 Atomscatm 36279 HLchlt 36366 LHypclh 37000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 |
This theorem is referenced by: cdleme21h 37350 |
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