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Theorem cdlemj1 30177
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 1094 . . . . . . . 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 5460 . . . . . . 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 5807 . . . . . 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 5808 . . . . 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 990 . . . . . 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 1095 . . . . . 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 994 . . . . . 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 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 ) ) )  ->  U  e.  E )
9 simp33 998 . . . . . 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 1096 . . . . . 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 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 ) ) )  ->  g  =/=  (  _I  |`  B ) )
12 simp31 996 . . . . . 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 2258 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
16 eqid 2258 . . . . . . 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 2258 . . . . . . 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 30176 . . . . . 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 1217 . . . . 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 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 ) ) )  ->  V  e.  E )
26 eqid 2258 . . . . . . 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 30176 . . . . . 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 1217 . . . . 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 2300 . . . 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 5807 . . 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 5808 . 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 1097 . . 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 993 . . 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 997 . . 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 2258 . . . 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 30176 . . 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 1217 . 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 2258 . . . 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 30176 . . 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 1217 . 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 2300 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997    _I cid 4276   `'ccnv 4660    |` cres 4663    o. ccom 4665   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   Atomscatm 28620   HLchlt 28707   LHypclh 29340   LTrncltrn 29457   trLctrl 29514   TEndoctendo 30108
This theorem is referenced by:  cdlemj2  30178
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-3or 940  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-rmo 2526  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-llines 28854  df-lplanes 28855  df-lvols 28856  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461  df-trl 29515  df-tendo 30111
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