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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 2731 | . 2 ⊢ ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) | |
9 | cdleme21g.d | . 2 ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) | |
10 | eqid 2731 | . 2 ⊢ ((𝑅 ∨ 𝑧) ∧ 𝑊) = ((𝑅 ∨ 𝑧) ∧ 𝑊) | |
11 | cdleme21g.n | . 2 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) | |
12 | eqid 2731 | . 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 39507 | 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 lecple 17209 joincjn 18269 meetcmee 18270 Atomscatm 38437 HLchlt 38524 LHypclh 39159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 |
This theorem is referenced by: cdleme21h 39509 |
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