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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme17d | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. We show, in their notation, fs(p)=q. TODO FIX COMMENT. (Contributed by NM, 11-Apr-2013.) |
Ref | Expression |
---|---|
cdlemef46.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemef46.l | ⊢ ≤ = (le‘𝐾) |
cdlemef46.j | ⊢ ∨ = (join‘𝐾) |
cdlemef46.m | ⊢ ∧ = (meet‘𝐾) |
cdlemef46.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemef46.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemef46.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdlemef46.d | ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) |
cdlemefs46.e | ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) |
cdlemef46.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) |
Ref | Expression |
---|---|
cdleme17d | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemef46.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemef46.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemef46.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemef46.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemef46.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemef46.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemef46.u | . . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
8 | cdlemef46.d | . . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) | |
9 | cdlemefs46.e | . . 3 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
10 | cdlemef46.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme17d4 39832 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → (𝐹‘𝑃) = 𝑄) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme17d3 39831 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝐹‘𝑃) = 𝑄) |
13 | 11, 12 | pm2.61dane 3028 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ⦋csb 3893 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 Basecbs 17151 lecple 17211 joincjn 18274 meetcmee 18275 Atomscatm 38597 HLchlt 38684 LHypclh 39319 |
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 ax-riotaBAD 38287 |
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-undef 8264 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-oposet 38510 df-ol 38512 df-oml 38513 df-covers 38600 df-ats 38601 df-atl 38632 df-cvlat 38656 df-hlat 38685 df-llines 38833 df-lplanes 38834 df-lvols 38835 df-lines 38836 df-psubsp 38838 df-pmap 38839 df-padd 39131 df-lhyp 39323 |
This theorem is referenced by: cdleme50ex 39894 cdlemg1a 39905 |
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