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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme19f | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 3. 𝐷, 𝐹, 𝑁, 𝑌, 𝐺, 𝑂 represent s2, f(s), fs(r), t2, f(t), ft(r). We prove that if r ≤ s ∨ t, then ft(r) = ft(r). (Contributed by NM, 14-Nov-2012.) |
Ref | Expression |
---|---|
cdleme19.l | ⊢ ≤ = (le‘𝐾) |
cdleme19.j | ⊢ ∨ = (join‘𝐾) |
cdleme19.m | ⊢ ∧ = (meet‘𝐾) |
cdleme19.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme19.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme19.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme19.f | ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) |
cdleme19.g | ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) |
cdleme19.d | ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) |
cdleme19.y | ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) |
cdleme19.n | ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) |
cdleme19.o | ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) |
Ref | Expression |
---|---|
cdleme19f | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → 𝑁 = 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme19.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme19.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme19.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleme19.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdleme19.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdleme19.u | . . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
7 | cdleme19.f | . . . 4 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) | |
8 | cdleme19.g | . . . 4 ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊))) | |
9 | cdleme19.d | . . . 4 ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊) | |
10 | cdleme19.y | . . . 4 ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme19e 40006 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → (𝐹 ∨ 𝐷) = (𝐺 ∨ 𝑌)) |
12 | 11 | oveq2d 7440 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌))) |
13 | cdleme19.n | . 2 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝐷)) | |
14 | cdleme19.o | . 2 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ 𝑌)) | |
15 | 12, 13, 14 | 3eqtr4g 2791 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑆 ∨ 𝑇)))) → 𝑁 = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 lecple 17273 joincjn 18336 meetcmee 18337 Atomscatm 38961 HLchlt 39048 LHypclh 39683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 |
This theorem is referenced by: cdleme20 40023 |
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