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Theorem cdlemd4 29557
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
Hypotheses
Ref Expression
cdlemd4.l  |-  .<_  =  ( le `  K )
cdlemd4.j  |-  .\/  =  ( join `  K )
cdlemd4.a  |-  A  =  ( Atoms `  K )
cdlemd4.h  |-  H  =  ( LHyp `  K
)
cdlemd4.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemd4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )

Proof of Theorem cdlemd4
StepHypRef Expression
1 simp11l 1071 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  K  e.  HL )
2 simp11r 1072 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  W  e.  H
)
3 simp21 993 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp22 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
5 simp231 1104 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  =/=  Q
)
6 cdlemd4.l . . . 4  |-  .<_  =  ( le `  K )
7 cdlemd4.j . . . 4  |-  .\/  =  ( join `  K )
8 cdlemd4.a . . . 4  |-  A  =  ( Atoms `  K )
9 cdlemd4.h . . . 4  |-  H  =  ( LHyp `  K
)
106, 7, 8, 9cdlemb2 29397 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/= 
Q )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) )
111, 2, 3, 4, 5, 10syl221anc 1198 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) ) )
12 simpl11 1035 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simpl12 1036 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( F  e.  T  /\  G  e.  T )
)
14 simpl13 1037 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  R  e.  A )
15 simpl21 1038 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
16 simprl 735 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  s  e.  A )
17 simprrl 743 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  -.  s  .<_  W )
1816, 17jca 520 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  (
s  e.  A  /\  -.  s  .<_  W ) )
19 hllat 28720 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
201, 19syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  K  e.  Lat )
2120adantr 453 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  K  e.  Lat )
22 eqid 2258 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2322, 8atbase 28646 . . . . . . . 8  |-  ( s  e.  A  ->  s  e.  ( Base `  K
) )
2423ad2antrl 711 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  s  e.  ( Base `  K
) )
25 simp21l 1077 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  e.  A
)
2622, 8atbase 28646 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2725, 26syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  e.  (
Base `  K )
)
2827adantr 453 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  P  e.  ( Base `  K
) )
29 simp22l 1079 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  Q  e.  A
)
3022, 8atbase 28646 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3129, 30syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  Q  e.  (
Base `  K )
)
3231adantr 453 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  Q  e.  ( Base `  K
) )
33 simprrr 744 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  -.  s  .<_  ( P  .\/  Q ) )
3422, 6, 7latnlej1l 14137 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( s  e.  (
Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K ) )  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  s  =/=  P
)
3534necomd 2504 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( s  e.  (
Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K ) )  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  P  =/=  s
)
3621, 24, 28, 32, 33, 35syl131anc 1200 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  P  =/=  s )
37 simpl22 1039 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
38 simpl23 1040 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )
396, 7, 8, 9cdlemd3 29556 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q )  /\  R  =/=  P
) )  /\  ( R  e.  A  /\  s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q ) ) )  ->  -.  R  .<_  ( P  .\/  s
) )
4012, 15, 37, 38, 14, 16, 33, 39syl133anc 1210 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  -.  R  .<_  ( P  .\/  s ) )
4136, 40jca 520 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( P  =/=  s  /\  -.  R  .<_  ( P  .\/  s ) ) )
42 simpl3l 1015 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( F `  P )  =  ( G `  P ) )
435adantr 453 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  P  =/=  Q )
4443, 33jca 520 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( P  =/=  Q  /\  -.  s  .<_  ( P  .\/  Q ) ) )
45 simpl3 965 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )
46 cdlemd4.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
476, 7, 8, 9, 46cdlemd2 29555 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  s  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  s  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  s )  =  ( G `  s ) )
4812, 13, 16, 15, 37, 44, 45, 47syl331anc 1212 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( F `  s )  =  ( G `  s ) )
496, 7, 8, 9, 46cdlemd2 29555 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( P  =/=  s  /\  -.  R  .<_  ( P  .\/  s ) ) )  /\  ( ( F `
 P )  =  ( G `  P
)  /\  ( F `  s )  =  ( G `  s ) ) )  ->  ( F `  R )  =  ( G `  R ) )
5012, 13, 14, 15, 18, 41, 42, 48, 49syl332anc 1218 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( F `  R )  =  ( G `  R ) )
5150exp32 591 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( s  e.  A  ->  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  ( F `  R )  =  ( G `  R ) ) ) )
5251rexlimdv 2641 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( E. s  e.  A  ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  ( F `  R )  =  ( G `  R ) ) )
5311, 52mpd 16 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )  /\  (
( F `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   Latclat 14113   Atomscatm 28620   HLchlt 28707   LHypclh 29340   LTrncltrn 29457
This theorem is referenced by:  cdlemd5  29558
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461
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