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Theorem cdleme17c 30770
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  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  ( Q  .\/  C ) )  =  Q )

Proof of Theorem cdleme17c
StepHypRef Expression
1 simp1l 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
2 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
3 simp31 993 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
4 cdleme17.j . . . . 5  |-  .\/  =  ( join `  K )
5 cdleme17.a . . . . 5  |-  A  =  ( Atoms `  K )
64, 5hlatjcom 29850 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
71, 2, 3, 6syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P ) )
87oveq1d 6055 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  ( Q  .\/  C ) )  =  ( ( Q  .\/  P ) 
./\  ( Q  .\/  C ) ) )
9 simp1r 982 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  W  e.  H )
10 simp2r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  P  .<_  W )
11 simp32 994 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  S  e.  A )
12 hllat 29846 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
131, 12syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  Lat )
14 eqid 2404 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
1514, 5atbase 29772 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1611, 15syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  S  e.  ( Base `  K )
)
1714, 5atbase 29772 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
182, 17syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  ( Base `  K )
)
1914, 5atbase 29772 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
203, 19syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  ( Base `  K )
)
21 simp33 995 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q
) )
22 cdleme17.l . . . . . . 7  |-  .<_  =  ( le `  K )
2314, 22, 4latnlej1l 14453 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  S  =/=  P )
2423necomd 2650 . . . . 5  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  P  =/=  S )
2513, 16, 18, 20, 21, 24syl131anc 1197 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  =/=  S )
26 cdleme17.m . . . . 5  |-  ./\  =  ( meet `  K )
27 cdleme17.h . . . . 5  |-  H  =  ( LHyp `  K
)
28 cdleme17.c . . . . 5  |-  C  =  ( ( P  .\/  S )  ./\  W )
2922, 4, 26, 5, 27, 28cdleme9a 30733 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( S  e.  A  /\  P  =/=  S ) )  ->  C  e.  A
)
301, 9, 2, 10, 11, 25, 29syl222anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  C  e.  A )
31 cdleme17.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
32 cdleme17.f . . . 4  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
33 cdleme17.g . . . 4  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( P  .\/  S )  ./\  W )
) )
3422, 4, 26, 5, 27, 31, 32, 33, 28cdleme17b 30769 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  C  .<_  ( P  .\/  Q
) )
3522, 4, 26, 52llnma1 30269 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  C  e.  A
)  /\  -.  C  .<_  ( P  .\/  Q
) )  ->  (
( Q  .\/  P
)  ./\  ( Q  .\/  C ) )  =  Q )
361, 2, 3, 30, 34, 35syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( Q  .\/  P )  ./\  ( Q  .\/  C ) )  =  Q )
378, 36eqtrd 2436 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  Q )  ./\  ( Q  .\/  C ) )  =  Q )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme17d1  30771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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