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Theorem cdleme38m 30945
Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on  P  .\/  Q line. TODO: FIX COMMENT (Contributed by NM, 13-Mar-2013.)
Hypotheses
Ref Expression
cdleme38.l  |-  .<_  =  ( le `  K )
cdleme38.j  |-  .\/  =  ( join `  K )
cdleme38.m  |-  ./\  =  ( meet `  K )
cdleme38.a  |-  A  =  ( Atoms `  K )
cdleme38.h  |-  H  =  ( LHyp `  K
)
cdleme38.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme38.e  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme38.d  |-  D  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )
cdleme38.v  |-  V  =  ( ( t  .\/  E )  ./\  W )
cdleme38.x  |-  X  =  ( ( u  .\/  D )  ./\  W )
cdleme38.f  |-  F  =  ( ( R  .\/  V )  ./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) )
cdleme38.g  |-  G  =  ( ( S  .\/  X )  ./\  ( D  .\/  ( ( u  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme38m  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  R  =  S )

Proof of Theorem cdleme38m
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp2 958 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )
3 simp311 1104 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  R  .<_  ( P  .\/  Q ) )
4 simp312 1105 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  S  .<_  ( P  .\/  Q ) )
5 simp313 1106 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  F  =  G )
63, 4jca 519 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( R  .<_  ( P 
.\/  Q )  /\  S  .<_  ( P  .\/  Q ) ) )
7 simp32 994 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) ) )
8 simp33 995 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) )
9 cdleme38.l . . . . . 6  |-  .<_  =  ( le `  K )
10 cdleme38.j . . . . . 6  |-  .\/  =  ( join `  K )
11 cdleme38.m . . . . . 6  |-  ./\  =  ( meet `  K )
12 cdleme38.a . . . . . 6  |-  A  =  ( Atoms `  K )
13 cdleme38.h . . . . . 6  |-  H  =  ( LHyp `  K
)
14 cdleme38.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
15 cdleme38.e . . . . . 6  |-  E  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
16 cdleme38.d . . . . . 6  |-  D  =  ( ( u  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  u )  ./\  W
) ) )
17 cdleme38.v . . . . . 6  |-  V  =  ( ( t  .\/  E )  ./\  W )
18 cdleme38.x . . . . . 6  |-  X  =  ( ( u  .\/  D )  ./\  W )
19 eqid 2404 . . . . . 6  |-  ( ( S  .\/  V ) 
./\  ( E  .\/  ( ( t  .\/  S )  ./\  W )
) )  =  ( ( S  .\/  V
)  ./\  ( E  .\/  ( ( t  .\/  S )  ./\  W )
) )
20 cdleme38.g . . . . . 6  |-  G  =  ( ( S  .\/  X )  ./\  ( D  .\/  ( ( u  .\/  S )  ./\  W )
) )
219, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20cdleme37m 30944 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q ) )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P 
.\/  Q ) )  /\  ( ( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P 
.\/  Q ) ) ) )  ->  (
( S  .\/  V
)  ./\  ( E  .\/  ( ( t  .\/  S )  ./\  W )
) )  =  G )
221, 2, 6, 7, 8, 21syl113anc 1196 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( ( S  .\/  V )  ./\  ( E  .\/  ( ( t  .\/  S )  ./\  W )
) )  =  G )
235, 22eqtr4d 2439 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  F  =  ( ( S  .\/  V )  ./\  ( E  .\/  ( ( t  .\/  S ) 
./\  W ) ) ) )
243, 4, 233jca 1134 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  -> 
( R  .<_  ( P 
.\/  Q )  /\  S  .<_  ( P  .\/  Q )  /\  F  =  ( ( S  .\/  V )  ./\  ( E  .\/  ( ( t  .\/  S )  ./\  W )
) ) ) )
25 eqid 2404 . . 3  |-  ( Base `  K )  =  (
Base `  K )
26 cdleme38.f . . 3  |-  F  =  ( ( R  .\/  V )  ./\  ( E  .\/  ( ( t  .\/  R )  ./\  W )
) )
2725, 9, 10, 11, 12, 13, 14, 15, 17, 26, 19cdleme36m 30943 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  ( ( S  .\/  V )  ./\  ( E  .\/  ( ( t  .\/  S ) 
./\  W ) ) ) )  /\  (
( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) ) ) )  ->  R  =  S )
281, 2, 24, 7, 27syl112anc 1188 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( ( R  .<_  ( P  .\/  Q )  /\  S  .<_  ( P 
.\/  Q )  /\  F  =  G )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) )  /\  (
( u  e.  A  /\  -.  u  .<_  W )  /\  -.  u  .<_  ( P  .\/  Q ) ) ) )  ->  R  =  S )
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   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme38n  30946
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-3or 937  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-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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