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Theorem cdlemj1 30277
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 5487 . . . . . . 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 5834 . . . . . 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 5835 . . . . 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 2284 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
16 eqid 2284 . . . . . . 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 2284 . . . . . . 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 30276 . . . . . 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 2284 . . . . . . 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 30276 . . . . . 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 2326 . . . 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 5834 . . 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 5835 . 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 2284 . . . 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 30276 . . 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 2284 . . . 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 30276 . . 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 2326 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 1628    e. wcel 1688    =/= wne 2447   class class class wbr 4024    _I cid 4303   `'ccnv 4687    |` cres 4690    o. ccom 4692   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614   TEndoctendo 30208
This theorem is referenced by:  cdlemj2  30278
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615  df-tendo 30211
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