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Theorem cdleme38m 31358
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 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 ) ) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp2 959 . 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 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 ) ) ) )  ->  R  .<_  ( P  .\/  Q ) )
4 simp312 1106 . . 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 1107 . . . 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 520 . . . . 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 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 ) ) ) )  -> 
( ( t  e.  A  /\  -.  t  .<_  W )  /\  -.  t  .<_  ( P  .\/  Q ) ) )
8 simp33 996 . . . . 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 2442 . . . . . 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 31357 . . . . 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 1197 . . . 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 2477 . . 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 1135 . 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 2442 . . 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 31356 . 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 1189 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   joincjn 14432   meetcmee 14433   Atomscatm 30159   HLchlt 30246   LHypclh 30879
This theorem is referenced by:  cdleme38n  31359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-llines 30393  df-lplanes 30394  df-lvols 30395  df-lines 30396  df-psubsp 30398  df-pmap 30399  df-padd 30691  df-lhyp 30883
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