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Theorem cdlemj1 31457
Description: Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
Hypotheses
Ref Expression
cdlemj.b  |-  B  =  ( Base `  K
)
cdlemj.h  |-  H  =  ( LHyp `  K
)
cdlemj.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemj.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemj.e  |-  E  =  ( ( TEndo `  K
) `  W )
cdlemj.l  |-  .<_  =  ( le `  K )
cdlemj.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdlemj1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( U `  h
) `  p )  =  ( ( V `
 h ) `  p ) )

Proof of Theorem cdlemj1
StepHypRef Expression
1 simp123 1091 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  ( U `  F )  =  ( V `  F ) )
21fveq1d 5721 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( U `  F
) `  p )  =  ( ( V `
 F ) `  p ) )
32oveq1d 6087 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( ( U `  F ) `  p
) ( join `  K
) ( R `  ( g  o.  `' F ) ) )  =  ( ( ( V `  F ) `
 p ) (
join `  K )
( R `  (
g  o.  `' F
) ) ) )
43oveq2d 6088 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( p ( join `  K ) ( R `
 g ) ) ( meet `  K
) ( ( ( U `  F ) `
 p ) (
join `  K )
( R `  (
g  o.  `' F
) ) ) )  =  ( ( p ( join `  K
) ( R `  g ) ) (
meet `  K )
( ( ( V `
 F ) `  p ) ( join `  K ) ( R `
 ( g  o.  `' F ) ) ) ) )
5 simp11 987 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simp131 1092 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  F  e.  T )
7 simp22 991 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  g  e.  T )
8 simp121 1089 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  U  e.  E )
9 simp33 995 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
p  e.  A  /\  -.  p  .<_  W ) )
10 simp132 1093 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  F  =/=  (  _I  |`  B ) )
11 simp23 992 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  g  =/=  (  _I  |`  B ) )
12 simp31 993 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  ( R `  F )  =/=  ( R `  g
) )
13 cdlemj.b . . . . . . 7  |-  B  =  ( Base `  K
)
14 cdlemj.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 eqid 2435 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
16 eqid 2435 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
17 cdlemj.a . . . . . . 7  |-  A  =  ( Atoms `  K )
18 cdlemj.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
19 cdlemj.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
20 cdlemj.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
21 cdlemj.e . . . . . . 7  |-  E  =  ( ( TEndo `  K
) `  W )
22 eqid 2435 . . . . . . 7  |-  ( ( p ( join `  K
) ( R `  g ) ) (
meet `  K )
( ( ( U `
 F ) `  p ) ( join `  K ) ( R `
 ( g  o.  `' F ) ) ) )  =  ( ( p ( join `  K
) ( R `  g ) ) (
meet `  K )
( ( ( U `
 F ) `  p ) ( join `  K ) ( R `
 ( g  o.  `' F ) ) ) )
2313, 14, 15, 16, 17, 18, 19, 20, 21, 22cdlemi 31456 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  g  e.  T )  /\  ( U  e.  E  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( F  =/=  (  _I  |`  B )  /\  g  =/=  (  _I  |`  B )  /\  ( R `  F )  =/=  ( R `  g ) ) )  ->  ( ( U `
 g ) `  p )  =  ( ( p ( join `  K ) ( R `
 g ) ) ( meet `  K
) ( ( ( U `  F ) `
 p ) (
join `  K )
( R `  (
g  o.  `' F
) ) ) ) )
245, 6, 7, 8, 9, 10, 11, 12, 23syl323anc 1214 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( U `  g
) `  p )  =  ( ( p ( join `  K
) ( R `  g ) ) (
meet `  K )
( ( ( U `
 F ) `  p ) ( join `  K ) ( R `
 ( g  o.  `' F ) ) ) ) )
25 simp122 1090 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  V  e.  E )
26 eqid 2435 . . . . . . 7  |-  ( ( p ( join `  K
) ( R `  g ) ) (
meet `  K )
( ( ( V `
 F ) `  p ) ( join `  K ) ( R `
 ( g  o.  `' F ) ) ) )  =  ( ( p ( join `  K
) ( R `  g ) ) (
meet `  K )
( ( ( V `
 F ) `  p ) ( join `  K ) ( R `
 ( g  o.  `' F ) ) ) )
2713, 14, 15, 16, 17, 18, 19, 20, 21, 26cdlemi 31456 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  g  e.  T )  /\  ( V  e.  E  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( F  =/=  (  _I  |`  B )  /\  g  =/=  (  _I  |`  B )  /\  ( R `  F )  =/=  ( R `  g ) ) )  ->  ( ( V `
 g ) `  p )  =  ( ( p ( join `  K ) ( R `
 g ) ) ( meet `  K
) ( ( ( V `  F ) `
 p ) (
join `  K )
( R `  (
g  o.  `' F
) ) ) ) )
285, 6, 7, 25, 9, 10, 11, 12, 27syl323anc 1214 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( V `  g
) `  p )  =  ( ( p ( join `  K
) ( R `  g ) ) (
meet `  K )
( ( ( V `
 F ) `  p ) ( join `  K ) ( R `
 ( g  o.  `' F ) ) ) ) )
294, 24, 283eqtr4d 2477 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( U `  g
) `  p )  =  ( ( V `
 g ) `  p ) )
3029oveq1d 6087 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( ( U `  g ) `  p
) ( join `  K
) ( R `  ( h  o.  `' g ) ) )  =  ( ( ( V `  g ) `
 p ) (
join `  K )
( R `  (
h  o.  `' g ) ) ) )
3130oveq2d 6088 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( p ( join `  K ) ( R `
 h ) ) ( meet `  K
) ( ( ( U `  g ) `
 p ) (
join `  K )
( R `  (
h  o.  `' g ) ) ) )  =  ( ( p ( join `  K
) ( R `  h ) ) (
meet `  K )
( ( ( V `
 g ) `  p ) ( join `  K ) ( R `
 ( h  o.  `' g ) ) ) ) )
32 simp133 1094 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  h  e.  T )
33 simp21 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  h  =/=  (  _I  |`  B ) )
34 simp32 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  ( R `  g )  =/=  ( R `  h
) )
35 eqid 2435 . . . 4  |-  ( ( p ( join `  K
) ( R `  h ) ) (
meet `  K )
( ( ( U `
 g ) `  p ) ( join `  K ) ( R `
 ( h  o.  `' g ) ) ) )  =  ( ( p ( join `  K ) ( R `
 h ) ) ( meet `  K
) ( ( ( U `  g ) `
 p ) (
join `  K )
( R `  (
h  o.  `' g ) ) ) )
3613, 14, 15, 16, 17, 18, 19, 20, 21, 35cdlemi 31456 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  h  e.  T )  /\  ( U  e.  E  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( g  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  g )  =/=  ( R `  h ) ) )  ->  ( ( U `
 h ) `  p )  =  ( ( p ( join `  K ) ( R `
 h ) ) ( meet `  K
) ( ( ( U `  g ) `
 p ) (
join `  K )
( R `  (
h  o.  `' g ) ) ) ) )
375, 7, 32, 8, 9, 11, 33, 34, 36syl323anc 1214 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( U `  h
) `  p )  =  ( ( p ( join `  K
) ( R `  h ) ) (
meet `  K )
( ( ( U `
 g ) `  p ) ( join `  K ) ( R `
 ( h  o.  `' g ) ) ) ) )
38 eqid 2435 . . . 4  |-  ( ( p ( join `  K
) ( R `  h ) ) (
meet `  K )
( ( ( V `
 g ) `  p ) ( join `  K ) ( R `
 ( h  o.  `' g ) ) ) )  =  ( ( p ( join `  K ) ( R `
 h ) ) ( meet `  K
) ( ( ( V `  g ) `
 p ) (
join `  K )
( R `  (
h  o.  `' g ) ) ) )
3913, 14, 15, 16, 17, 18, 19, 20, 21, 38cdlemi 31456 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  h  e.  T )  /\  ( V  e.  E  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( g  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  g )  =/=  ( R `  h ) ) )  ->  ( ( V `
 h ) `  p )  =  ( ( p ( join `  K ) ( R `
 h ) ) ( meet `  K
) ( ( ( V `  g ) `
 p ) (
join `  K )
( R `  (
h  o.  `' g ) ) ) ) )
405, 7, 32, 25, 9, 11, 33, 34, 39syl323anc 1214 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( V `  h
) `  p )  =  ( ( p ( join `  K
) ( R `  h ) ) (
meet `  K )
( ( ( V `
 g ) `  p ) ( join `  K ) ( R `
 ( h  o.  `' g ) ) ) ) )
4131, 37, 403eqtr4d 2477 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  V  e.  E  /\  ( U `  F )  =  ( V `  F ) )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  h  e.  T )
)  /\  ( h  =/=  (  _I  |`  B )  /\  g  e.  T  /\  g  =/=  (  _I  |`  B ) )  /\  ( ( R `
 F )  =/=  ( R `  g
)  /\  ( R `  g )  =/=  ( R `  h )  /\  ( p  e.  A  /\  -.  p  .<_  W ) ) )  ->  (
( U `  h
) `  p )  =  ( ( V `
 h ) `  p ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204    _I cid 4485   `'ccnv 4868    |` cres 4871    o. ccom 4873   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737   trLctrl 30794   TEndoctendo 31388
This theorem is referenced by:  cdlemj2  31458
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 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-rmo 2705  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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795  df-tendo 31391
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