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Theorem cdlemd4 30935
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
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 simp11l 1068 . . 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 1069 . . 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 990 . . 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 991 . . 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 1101 . . 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 30775 . . 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 1195 . 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 1032 . . 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 ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simpl12 1033 . . 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  e.  T  /\  G  e.  T )
)
14 simpl13 1034 . . 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 ) ) ) )  ->  R  e.  A )
15 simpl21 1035 . . 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 ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
16 simprl 733 . . . 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 ) ) ) )  ->  s  e.  A )
17 simprrl 741 . . . 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 ) ) ) )  ->  -.  s  .<_  W )
1816, 17jca 519 . . 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 ) ) ) )  ->  (
s  e.  A  /\  -.  s  .<_  W ) )
19 hllat 30098 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
201, 19syl 16 . . . . . 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 ) ) )  ->  K  e.  Lat )
2120adantr 452 . . . . 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.  Lat )
22 eqid 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2322, 8atbase 30024 . . . . . 6  |-  ( s  e.  A  ->  s  e.  ( Base `  K
) )
2423ad2antrl 709 . . . . 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.  ( Base `  K
) )
25 simp21l 1074 . . . . . . 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 ) ) )  ->  P  e.  A
)
2622, 8atbase 30024 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2725, 26syl 16 . . . . . 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 ) ) )  ->  P  e.  (
Base `  K )
)
2827adantr 452 . . . . 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.  ( Base `  K
) )
29 simp22l 1076 . . . . . . 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 ) ) )  ->  Q  e.  A
)
3022, 8atbase 30024 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3129, 30syl 16 . . . . . 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 ) ) )  ->  Q  e.  (
Base `  K )
)
3231adantr 452 . . . . 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 ) ) ) )  ->  Q  e.  ( Base `  K
) )
33 simprrr 742 . . . . 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  .<_  ( P  .\/  Q ) )
3422, 6, 7latnlej1l 14490 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( s  e.  (
Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K ) )  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  s  =/=  P
)
3534necomd 2681 . . . . 5  |-  ( ( 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 1197 . . . 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 ) ) ) )  ->  P  =/=  s )
37 simpl22 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 ) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
38 simpl23 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 ) ) ) )  ->  ( P  =/=  Q  /\  R  .<_  ( P  .\/  Q
)  /\  R  =/=  P ) )
396, 7, 8, 9cdlemd3 30934 . . . . 5  |-  ( ( ( 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 1207 . . . 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 ) ) ) )  ->  -.  R  .<_  ( P  .\/  s ) )
4136, 40jca 519 . . 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 ) ) ) )  ->  ( P  =/=  s  /\  -.  R  .<_  ( P  .\/  s ) ) )
42 simpl3l 1012 . . 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 `  P )  =  ( G `  P ) )
435adantr 452 . . . . 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  =/=  Q )
4443, 33jca 519 . . . 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 ) ) ) )  ->  ( P  =/=  Q  /\  -.  s  .<_  ( P  .\/  Q ) ) )
45 simpl3 962 . . . 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 `  P
)  =  ( G `
 P )  /\  ( F `  Q )  =  ( G `  Q ) ) )
46 cdlemd4.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
476, 7, 8, 9, 46cdlemd2 30933 . . . 4  |-  ( ( ( ( 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 1209 . . 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 `  s )  =  ( G `  s ) )
496, 7, 8, 9, 46cdlemd2 30933 . . 3  |-  ( ( ( ( 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 1215 . 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 ) ) )  /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) ) ) )  ->  ( F `  R )  =  ( G `  R ) )
5111, 50rexlimddv 2826 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 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29998   HLchlt 30085   LHypclh 30718   LTrncltrn 30835
This theorem is referenced by:  cdlemd5  30936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839
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