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Theorem cdleme20k 30801
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 20-Nov-2012.)
Hypotheses
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 )
cdleme20.v  |-  V  =  ( ( S  .\/  T )  ./\  W )
Assertion
Ref Expression
cdleme20k  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( F  .\/  D )  =/=  ( P  .\/  Q ) )

Proof of Theorem cdleme20k
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
3 simp13 989 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  Q  e.  A )
4 simp2r 984 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
5 simp2l 983 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
6 simp3r 986 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  R  .<_  ( P  .\/  Q ) )
7 simp3l 985 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q
) )
8 cdleme19.l . . . 4  |-  .<_  =  ( le `  K )
9 cdleme19.j . . . 4  |-  .\/  =  ( join `  K )
10 cdleme19.m . . . 4  |-  ./\  =  ( meet `  K )
11 cdleme19.a . . . 4  |-  A  =  ( Atoms `  K )
12 cdleme19.h . . . 4  |-  H  =  ( LHyp `  K
)
13 cdleme19.d . . . 4  |-  D  =  ( ( R  .\/  S )  ./\  W )
148, 9, 10, 11, 12, 13cdlemednpq 30781 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  D  .<_  ( P 
.\/  Q ) )
151, 2, 3, 4, 5, 6, 7, 14syl133anc 1207 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  -.  D  .<_  ( P  .\/  Q
) )
16 simp11l 1068 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  HL )
17 hllat 29846 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
1816, 17syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  K  e.  Lat )
19 simp11r 1069 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  W  e.  H )
20 simp2ll 1024 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  S  e.  A )
21 cdleme19.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
22 cdleme19.f . . . . . . 7  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
23 eqid 2404 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
248, 9, 10, 11, 12, 21, 22, 23cdleme1b 30708 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A ) )  ->  F  e.  ( Base `  K ) )
2516, 19, 2, 3, 20, 24syl23anc 1191 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  F  e.  ( Base `  K )
)
26 simp2rl 1026 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  R  e.  A )
278, 9, 10, 11, 12, 13, 23cdlemedb 30779 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  D  e.  ( Base `  K ) )
2816, 19, 26, 20, 27syl22anc 1185 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  D  e.  ( Base `  K )
)
2923, 8, 9latlej2 14445 . . . . 5  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  D  e.  ( Base `  K
) )  ->  D  .<_  ( F  .\/  D
) )
3018, 25, 28, 29syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  D  .<_  ( F  .\/  D ) )
31 breq2 4176 . . . 4  |-  ( ( F  .\/  D )  =  ( P  .\/  Q )  ->  ( D  .<_  ( F  .\/  D
)  <->  D  .<_  ( P 
.\/  Q ) ) )
3230, 31syl5ibcom 212 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( ( F  .\/  D )  =  ( P  .\/  Q
)  ->  D  .<_  ( P  .\/  Q ) ) )
3332necon3bd 2604 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( -.  D  .<_  ( P  .\/  Q )  ->  ( F  .\/  D )  =/=  ( P  .\/  Q ) ) )
3415, 33mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
.\/  Q ) ) )  ->  ( F  .\/  D )  =/=  ( P  .\/  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:  cdleme20l  30804
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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