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Theorem cdlemd2 36321
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd2.l  |-  .<_  =  ( le `  K )
cdlemd2.j  |-  .\/  =  ( join `  K )
cdlemd2.a  |-  A  =  ( Atoms `  K )
cdlemd2.h  |-  H  =  ( LHyp `  K
)
cdlemd2.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemd2  |-  ( ( ( ( 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )

Proof of Theorem cdlemd2
StepHypRef Expression
1 simp3l 1022 . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  P )  =  ( G `  P ) )
2 simp11 1024 . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp12l 1107 . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  F  e.  T )
4 simp11l 1105 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  K  e.  HL )
5 hllat 35485 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 16 . . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  K  e.  Lat )
7 simp21l 1111 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  e.  A )
8 simp13 1026 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  R  e.  A )
9 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
10 cdlemd2.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
11 cdlemd2.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
129, 10, 11hlatjcl 35488 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
134, 7, 8, 12syl3anc 1226 . . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
14 simp11r 1106 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  W  e.  H )
15 cdlemd2.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
169, 15lhpbase 36119 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1714, 16syl 16 . . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  W  e.  ( Base `  K
) )
18 eqid 2454 . . . . . . . . . 10  |-  ( meet `  K )  =  (
meet `  K )
199, 18latmcl 15881 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R ) (
meet `  K ) W )  e.  (
Base `  K )
)
206, 13, 17, 19syl3anc 1226 . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( P  .\/  R
) ( meet `  K
) W )  e.  ( Base `  K
) )
21 cdlemd2.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
229, 21, 18latmle2 15906 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R ) (
meet `  K ) W )  .<_  W )
236, 13, 17, 22syl3anc 1226 . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( P  .\/  R
) ( meet `  K
) W )  .<_  W )
24 cdlemd2.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
259, 21, 15, 24ltrnval1 36255 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( ( P 
.\/  R ) (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( P  .\/  R ) ( meet `  K
) W )  .<_  W ) )  -> 
( F `  (
( P  .\/  R
) ( meet `  K
) W ) )  =  ( ( P 
.\/  R ) (
meet `  K ) W ) )
262, 3, 20, 23, 25syl112anc 1230 . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( P  .\/  R ) (
meet `  K ) W ) )  =  ( ( P  .\/  R ) ( meet `  K
) W ) )
27 simp12r 1108 . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  G  e.  T )
289, 21, 15, 24ltrnval1 36255 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( ( ( P 
.\/  R ) (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( P  .\/  R ) ( meet `  K
) W )  .<_  W ) )  -> 
( G `  (
( P  .\/  R
) ( meet `  K
) W ) )  =  ( ( P 
.\/  R ) (
meet `  K ) W ) )
292, 27, 20, 23, 28syl112anc 1230 . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( ( P  .\/  R ) (
meet `  K ) W ) )  =  ( ( P  .\/  R ) ( meet `  K
) W ) )
3026, 29eqtr4d 2498 . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( P  .\/  R ) (
meet `  K ) W ) )  =  ( G `  (
( P  .\/  R
) ( meet `  K
) W ) ) )
311, 30oveq12d 6288 . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( F `  P
)  .\/  ( F `  ( ( P  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( G `  P ) 
.\/  ( G `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
329, 11atbase 35411 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
337, 32syl 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  e.  ( Base `  K
) )
349, 10, 15, 24ltrnj 36253 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  (
Base `  K )  /\  ( ( P  .\/  R ) ( meet `  K
) W )  e.  ( Base `  K
) ) )  -> 
( F `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) )  =  ( ( F `
 P )  .\/  ( F `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
352, 3, 33, 20, 34syl112anc 1230 . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( F `  P ) 
.\/  ( F `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
369, 10, 15, 24ltrnj 36253 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  (
Base `  K )  /\  ( ( P  .\/  R ) ( meet `  K
) W )  e.  ( Base `  K
) ) )  -> 
( G `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) )  =  ( ( G `
 P )  .\/  ( G `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
372, 27, 33, 20, 36syl112anc 1230 . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( G `  P ) 
.\/  ( G `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
3831, 35, 373eqtr4d 2505 . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) )  =  ( G `
 ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
39 simp3r 1023 . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  Q )  =  ( G `  Q ) )
40 simp22l 1113 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  Q  e.  A )
419, 10, 11hlatjcl 35488 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
424, 40, 8, 41syl3anc 1226 . . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
439, 18latmcl 15881 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R ) (
meet `  K ) W )  e.  (
Base `  K )
)
446, 42, 17, 43syl3anc 1226 . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( Q  .\/  R
) ( meet `  K
) W )  e.  ( Base `  K
) )
459, 21, 18latmle2 15906 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R ) (
meet `  K ) W )  .<_  W )
466, 42, 17, 45syl3anc 1226 . . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( Q  .\/  R
) ( meet `  K
) W )  .<_  W )
479, 21, 15, 24ltrnval1 36255 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( ( Q 
.\/  R ) (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( Q  .\/  R ) ( meet `  K
) W )  .<_  W ) )  -> 
( F `  (
( Q  .\/  R
) ( meet `  K
) W ) )  =  ( ( Q 
.\/  R ) (
meet `  K ) W ) )
482, 3, 44, 46, 47syl112anc 1230 . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( Q  .\/  R ) (
meet `  K ) W ) )  =  ( ( Q  .\/  R ) ( meet `  K
) W ) )
499, 21, 15, 24ltrnval1 36255 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( ( ( Q 
.\/  R ) (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( Q  .\/  R ) ( meet `  K
) W )  .<_  W ) )  -> 
( G `  (
( Q  .\/  R
) ( meet `  K
) W ) )  =  ( ( Q 
.\/  R ) (
meet `  K ) W ) )
502, 27, 44, 46, 49syl112anc 1230 . . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( ( Q  .\/  R ) (
meet `  K ) W ) )  =  ( ( Q  .\/  R ) ( meet `  K
) W ) )
5148, 50eqtr4d 2498 . . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( Q  .\/  R ) (
meet `  K ) W ) )  =  ( G `  (
( Q  .\/  R
) ( meet `  K
) W ) ) )
5239, 51oveq12d 6288 . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( F `  Q
)  .\/  ( F `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( G `  Q ) 
.\/  ( G `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
539, 11atbase 35411 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
5440, 53syl 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  Q  e.  ( Base `  K
) )
559, 10, 15, 24ltrnj 36253 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( Q  e.  (
Base `  K )  /\  ( ( Q  .\/  R ) ( meet `  K
) W )  e.  ( Base `  K
) ) )  -> 
( F `  ( Q  .\/  ( ( Q 
.\/  R ) (
meet `  K ) W ) ) )  =  ( ( F `
 Q )  .\/  ( F `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
562, 3, 54, 44, 55syl112anc 1230 . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( F `  Q ) 
.\/  ( F `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
579, 10, 15, 24ltrnj 36253 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( Q  e.  (
Base `  K )  /\  ( ( Q  .\/  R ) ( meet `  K
) W )  e.  ( Base `  K
) ) )  -> 
( G `  ( Q  .\/  ( ( Q 
.\/  R ) (
meet `  K ) W ) ) )  =  ( ( G `
 Q )  .\/  ( G `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
582, 27, 54, 44, 57syl112anc 1230 . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( G `  Q ) 
.\/  ( G `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
5952, 56, 583eqtr4d 2505 . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) )  =  ( G `
 ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
6038, 59oveq12d 6288 . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( F `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )  =  ( ( G `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( G `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
619, 10latjcl 15880 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( P  .\/  R
) ( meet `  K
) W )  e.  ( Base `  K
) )  ->  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) )  e.  ( Base `  K
) )
626, 33, 20, 61syl3anc 1226 . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) )  e.  ( Base `  K
) )
639, 10latjcl 15880 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( Q  .\/  R
) ( meet `  K
) W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( Q 
.\/  R ) (
meet `  K ) W ) )  e.  ( Base `  K
) )
646, 54, 44, 63syl3anc 1226 . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( Q  .\/  ( ( Q 
.\/  R ) (
meet `  K ) W ) )  e.  ( Base `  K
) )
659, 18, 15, 24ltrnm 36252 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) )  e.  ( Base `  K
) ) )  -> 
( F `  (
( P  .\/  (
( P  .\/  R
) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )  =  ( ( F `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
662, 3, 62, 64, 65syl112anc 1230 . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) (
meet `  K )
( Q  .\/  (
( Q  .\/  R
) ( meet `  K
) W ) ) ) )  =  ( ( F `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
679, 18, 15, 24ltrnm 36252 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) )  e.  ( Base `  K
) ) )  -> 
( G `  (
( P  .\/  (
( P  .\/  R
) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )  =  ( ( G `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( G `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
682, 27, 62, 64, 67syl112anc 1230 . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) (
meet `  K )
( Q  .\/  (
( Q  .\/  R
) ( meet `  K
) W ) ) ) )  =  ( ( G `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( G `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
6960, 66, 683eqtr4d 2505 . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) (
meet `  K )
( Q  .\/  (
( Q  .\/  R
) ( meet `  K
) W ) ) ) )  =  ( G `  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
70 simp21 1027 . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
71 simp22 1028 . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
72 simp23l 1115 . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  =/=  Q )
73 simp23r 1116 . . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
748, 72, 733jca 1174 . . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( R  e.  A  /\  P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )
7521, 10, 18, 11, 15cdlemd1 36320 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  =  ( ( P  .\/  (
( P  .\/  R
) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
762, 70, 71, 74, 75syl13anc 1228 . . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  R  =  ( ( P 
.\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) (
meet `  K )
( Q  .\/  (
( Q  .\/  R
) ( meet `  K
) W ) ) ) )
7776fveq2d 5852 . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( F `  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
7876fveq2d 5852 . 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  R )  =  ( G `  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
7969, 77, 783eqtr4d 2505 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 ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Latclat 15874   Atomscatm 35385   HLchlt 35472   LHypclh 36105   LTrncltrn 36222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226
This theorem is referenced by:  cdlemd4  36323  cdlemd5  36324
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