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Theorem cdleme18b 30774
Description: Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph.  F,  G represent f(s), fs(q) respectively. We show  -. fs(q)  =/= q. (Contributed by NM, 12-Oct-2012.)
Hypotheses
Ref Expression
cdleme18.l  |-  .<_  =  ( le `  K )
cdleme18.j  |-  .\/  =  ( join `  K )
cdleme18.m  |-  ./\  =  ( meet `  K )
cdleme18.a  |-  A  =  ( Atoms `  K )
cdleme18.h  |-  H  =  ( LHyp `  K
)
cdleme18.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme18.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme18.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( Q  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme18b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  G  =/=  Q )

Proof of Theorem cdleme18b
StepHypRef Expression
1 eqid 2404 . 2  |-  Q  =  Q
2 oveq2 6048 . . . . . . 7  |-  ( G  =  Q  ->  ( Q  .\/  G )  =  ( Q  .\/  Q
) )
3 simp1l 981 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
4 simp22l 1076 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
5 cdleme18.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
6 cdleme18.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
75, 6hlatjidm 29851 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
83, 4, 7syl2anc 643 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( Q  .\/  Q )  =  Q )
92, 8sylan9eqr 2458 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  G  =  Q )  ->  ( Q  .\/  G
)  =  Q )
10 simp1 957 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simp21l 1074 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
12 simp22 991 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
13 simp23 992 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
14 cdleme18.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
1514, 5, 6hlatlej2 29858 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
163, 11, 4, 15syl3anc 1184 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  Q  .<_  ( P  .\/  Q ) )
17 cdleme18.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
18 cdleme18.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
19 cdleme18.u . . . . . . . . 9  |-  U  =  ( ( P  .\/  Q )  ./\  W )
20 cdleme18.f . . . . . . . . 9  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
21 cdleme18.g . . . . . . . . 9  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( Q  .\/  S )  ./\  W )
) )
2214, 5, 17, 6, 18, 19, 20, 21cdleme5 30722 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  Q  .<_  ( P  .\/  Q ) ) )  -> 
( Q  .\/  G
)  =  ( P 
.\/  Q ) )
2310, 11, 4, 12, 13, 16, 22syl132anc 1202 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( Q  .\/  G )  =  ( P  .\/  Q ) )
2423adantr 452 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  G  =  Q )  ->  ( Q  .\/  G
)  =  ( P 
.\/  Q ) )
259, 24eqtr3d 2438 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  G  =  Q )  ->  Q  =  ( P 
.\/  Q ) )
26 simp3l 985 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  P  =/=  Q )
275, 62atneat 29997 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  -.  ( P  .\/  Q )  e.  A )
283, 11, 4, 26, 27syl13anc 1186 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  ( P  .\/  Q )  e.  A )
29 nelne2 2657 . . . . . . . 8  |-  ( ( Q  e.  A  /\  -.  ( P  .\/  Q
)  e.  A )  ->  Q  =/=  ( P  .\/  Q ) )
3029necomd 2650 . . . . . . 7  |-  ( ( Q  e.  A  /\  -.  ( P  .\/  Q
)  e.  A )  ->  ( P  .\/  Q )  =/=  Q )
314, 28, 30syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  =/=  Q
)
3231adantr 452 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  G  =  Q )  ->  ( P  .\/  Q
)  =/=  Q )
3325, 32eqnetrd 2585 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  G  =  Q )  ->  Q  =/=  Q )
3433ex 424 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( G  =  Q  ->  Q  =/= 
Q ) )
3534necon2d 2617 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  ( Q  =  Q  ->  G  =/= 
Q ) )
361, 35mpi 17 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  G  =/=  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   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme18c  30775
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-llines 29980  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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