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Theorem cdleme17c 36955
Description: Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. 𝐶 represents s1. We show, in their notation, (p q) (q s1)=q. (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
cdleme17.l = (le‘𝐾)
cdleme17.j = (join‘𝐾)
cdleme17.m = (meet‘𝐾)
cdleme17.a 𝐴 = (Atoms‘𝐾)
cdleme17.h 𝐻 = (LHyp‘𝐾)
cdleme17.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme17.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme17.g 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))
cdleme17.c 𝐶 = ((𝑃 𝑆) 𝑊)
Assertion
Ref Expression
cdleme17c (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑃 𝑄) (𝑄 𝐶)) = 𝑄)

Proof of Theorem cdleme17c
StepHypRef Expression
1 simp1l 1190 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐾 ∈ HL)
2 simp2l 1192 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃𝐴)
3 simp31 1202 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑄𝐴)
4 cdleme17.j . . . . 5 = (join‘𝐾)
5 cdleme17.a . . . . 5 𝐴 = (Atoms‘𝐾)
64, 5hlatjcom 36035 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
71, 2, 3, 6syl3anc 1364 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑃 𝑄) = (𝑄 𝑃))
87oveq1d 7031 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑃 𝑄) (𝑄 𝐶)) = ((𝑄 𝑃) (𝑄 𝐶)))
9 simp1r 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑊𝐻)
10 simp2r 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑃 𝑊)
11 simp32 1203 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆𝐴)
121hllatd 36031 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐾 ∈ Lat)
13 eqid 2795 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1413, 5atbase 35956 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1511, 14syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑆 ∈ (Base‘𝐾))
1613, 5atbase 35956 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
172, 16syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃 ∈ (Base‘𝐾))
1813, 5atbase 35956 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
193, 18syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑄 ∈ (Base‘𝐾))
20 simp33 1204 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
21 cdleme17.l . . . . . . 7 = (le‘𝐾)
2213, 21, 4latnlej1l 17508 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑆𝑃)
2322necomd 3039 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝑆)
2412, 15, 17, 19, 20, 23syl131anc 1376 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑃𝑆)
25 cdleme17.m . . . . 5 = (meet‘𝐾)
26 cdleme17.h . . . . 5 𝐻 = (LHyp‘𝐾)
27 cdleme17.c . . . . 5 𝐶 = ((𝑃 𝑆) 𝑊)
2821, 4, 25, 5, 26, 27cdleme9a 36918 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑆𝐴𝑃𝑆)) → 𝐶𝐴)
291, 9, 2, 10, 11, 24, 28syl222anc 1379 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐶𝐴)
30 cdleme17.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
31 cdleme17.f . . . 4 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
32 cdleme17.g . . . 4 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))
3321, 4, 25, 5, 26, 30, 31, 32, 27cdleme17b 36954 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝐶 (𝑃 𝑄))
3421, 4, 25, 52llnma1 36454 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝐶𝐴) ∧ ¬ 𝐶 (𝑃 𝑄)) → ((𝑄 𝑃) (𝑄 𝐶)) = 𝑄)
351, 2, 3, 29, 33, 34syl131anc 1376 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑄 𝑃) (𝑄 𝐶)) = 𝑄)
368, 35eqtrd 2831 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝑃 𝑄) (𝑄 𝐶)) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984   class class class wbr 4962  cfv 6225  (class class class)co 7016  Basecbs 16312  lecple 16401  joincjn 17383  meetcmee 17384  Latclat 17484  Atomscatm 35930  HLchlt 36017  LHypclh 36651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-proset 17367  df-poset 17385  df-plt 17397  df-lub 17413  df-glb 17414  df-join 17415  df-meet 17416  df-p0 17478  df-p1 17479  df-lat 17485  df-clat 17547  df-oposet 35843  df-ol 35845  df-oml 35846  df-covers 35933  df-ats 35934  df-atl 35965  df-cvlat 35989  df-hlat 36018  df-psubsp 36170  df-pmap 36171  df-padd 36463  df-lhyp 36655
This theorem is referenced by:  cdleme17d1  36956
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