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Theorem cdlemg46 30054
Description: Part of proof of Lemma G of [Crawley] p. 116, seventh line of third paragraph on p. 117: "hf and f have different traces." (Contributed by NM, 5-Jun-2013.)
Hypotheses
Ref Expression
cdlemg46.b  |-  B  =  ( Base `  K
)
cdlemg46.h  |-  H  =  ( LHyp `  K
)
cdlemg46.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg46.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg46  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  ( h  o.  F
) )  =/=  ( R `  F )
)
Distinct variable groups:    h, F    h, H    h, K    R, h    T, h    h, W
Allowed substitution hint:    B( h)

Proof of Theorem cdlemg46
StepHypRef Expression
1 simpl1l 1011 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  K  e.  HL )
2 simp1 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp2r 987 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  h  e.  T
)
4 simp32 997 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  h  =/=  (  _I  |`  B ) )
5 cdlemg46.b . . . . . 6  |-  B  =  ( Base `  K
)
6 eqid 2256 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
7 cdlemg46.h . . . . . 6  |-  H  =  ( LHyp `  K
)
8 cdlemg46.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemg46.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
105, 6, 7, 8, 9trlnidat 29492 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  h  =/=  (  _I  |`  B ) )  ->  ( R `  h )  e.  (
Atoms `  K ) )
112, 3, 4, 10syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  h )  e.  (
Atoms `  K ) )
1211adantr 453 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  h )  e.  (
Atoms `  K ) )
13 simp2l 986 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  F  e.  T
)
14 simp31 996 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  F  =/=  (  _I  |`  B ) )
155, 6, 7, 8, 9trlnidat 29492 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  (
Atoms `  K ) )
162, 13, 14, 15syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  F )  e.  (
Atoms `  K ) )
1716adantr 453 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  F )  e.  (
Atoms `  K ) )
18 simpl33 1043 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  h )  =/=  ( R `  F )
)
19 simpr 449 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  ( h  o.  F
) )  e.  (
Atoms `  K ) )
207, 8ltrnco 30038 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T  /\  F  e.  T
)  ->  ( h  o.  F )  e.  T
)
212, 3, 13, 20syl3anc 1187 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( h  o.  F )  e.  T
)
227, 8ltrncnv 29465 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  `' F  e.  T )
232, 13, 22syl2anc 645 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  `' F  e.  T )
24 eqid 2256 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
25 eqid 2256 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
2624, 25, 7, 8, 9trlco 30046 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  o.  F )  e.  T  /\  `' F  e.  T
)  ->  ( R `  ( ( h  o.  F )  o.  `' F ) ) ( le `  K ) ( ( R `  ( h  o.  F
) ) ( join `  K ) ( R `
 `' F ) ) )
272, 21, 23, 26syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  ( ( h  o.  F )  o.  `' F ) ) ( le `  K ) ( ( R `  ( h  o.  F
) ) ( join `  K ) ( R `
 `' F ) ) )
28 coass 5143 . . . . . . . 8  |-  ( ( h  o.  F )  o.  `' F )  =  ( h  o.  ( F  o.  `' F ) )
295, 7, 8ltrn1o 29443 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
302, 13, 29syl2anc 645 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  F : B -1-1-onto-> B
)
31 f1ococnv2 5403 . . . . . . . . . . 11  |-  ( F : B -1-1-onto-> B  ->  ( F  o.  `' F )  =  (  _I  |`  B )
)
3230, 31syl 17 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( F  o.  `' F )  =  (  _I  |`  B )
)
3332coeq2d 4799 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( h  o.  ( F  o.  `' F ) )  =  ( h  o.  (  _I  |`  B ) ) )
345, 7, 8ltrn1o 29443 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  h  e.  T
)  ->  h : B
-1-1-onto-> B )
352, 3, 34syl2anc 645 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  h : B -1-1-onto-> B
)
36 f1of 5375 . . . . . . . . . 10  |-  ( h : B -1-1-onto-> B  ->  h : B
--> B )
37 fcoi1 5318 . . . . . . . . . 10  |-  ( h : B --> B  -> 
( h  o.  (  _I  |`  B ) )  =  h )
3835, 36, 373syl 20 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( h  o.  (  _I  |`  B ) )  =  h )
3933, 38eqtrd 2288 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( h  o.  ( F  o.  `' F ) )  =  h )
4028, 39syl5eq 2300 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( ( h  o.  F )  o.  `' F )  =  h )
4140fveq2d 5427 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  ( ( h  o.  F )  o.  `' F ) )  =  ( R `  h
) )
427, 8, 9trlcnv 29484 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  `' F )  =  ( R `  F ) )
432, 13, 42syl2anc 645 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  `' F )  =  ( R `  F ) )
4443oveq2d 5773 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( ( R `
 ( h  o.  F ) ) (
join `  K )
( R `  `' F ) )  =  ( ( R `  ( h  o.  F
) ) ( join `  K ) ( R `
 F ) ) )
4527, 41, 443brtr3d 3992 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  h ) ( le
`  K ) ( ( R `  (
h  o.  F ) ) ( join `  K
) ( R `  F ) ) )
4645adantr 453 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  h ) ( le
`  K ) ( ( R `  (
h  o.  F ) ) ( join `  K
) ( R `  F ) ) )
4724, 25, 6hlatlej2 28695 . . . . 5  |-  ( ( K  e.  HL  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K
)  /\  ( R `  F )  e.  (
Atoms `  K ) )  ->  ( R `  F ) ( le
`  K ) ( ( R `  (
h  o.  F ) ) ( join `  K
) ( R `  F ) ) )
481, 19, 17, 47syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  F ) ( le
`  K ) ( ( R `  (
h  o.  F ) ) ( join `  K
) ( R `  F ) ) )
49 hllat 28683 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
501, 49syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  K  e.  Lat )
515, 6atbase 28609 . . . . . 6  |-  ( ( R `  h )  e.  ( Atoms `  K
)  ->  ( R `  h )  e.  B
)
5212, 51syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  h )  e.  B
)
535, 6atbase 28609 . . . . . 6  |-  ( ( R `  F )  e.  ( Atoms `  K
)  ->  ( R `  F )  e.  B
)
5417, 53syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  F )  e.  B
)
555, 25, 6hlatjcl 28686 . . . . . 6  |-  ( ( K  e.  HL  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K
)  /\  ( R `  F )  e.  (
Atoms `  K ) )  ->  ( ( R `
 ( h  o.  F ) ) (
join `  K )
( R `  F
) )  e.  B
)
561, 19, 17, 55syl3anc 1187 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( ( R `  ( h  o.  F ) ) (
join `  K )
( R `  F
) )  e.  B
)
575, 24, 25latjle12 14095 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( R `  h )  e.  B  /\  ( R `  F
)  e.  B  /\  ( ( R `  ( h  o.  F
) ) ( join `  K ) ( R `
 F ) )  e.  B ) )  ->  ( ( ( R `  h ) ( le `  K
) ( ( R `
 ( h  o.  F ) ) (
join `  K )
( R `  F
) )  /\  ( R `  F )
( le `  K
) ( ( R `
 ( h  o.  F ) ) (
join `  K )
( R `  F
) ) )  <->  ( ( R `  h )
( join `  K )
( R `  F
) ) ( le
`  K ) ( ( R `  (
h  o.  F ) ) ( join `  K
) ( R `  F ) ) ) )
5850, 52, 54, 56, 57syl13anc 1189 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( (
( R `  h
) ( le `  K ) ( ( R `  ( h  o.  F ) ) ( join `  K
) ( R `  F ) )  /\  ( R `  F ) ( le `  K
) ( ( R `
 ( h  o.  F ) ) (
join `  K )
( R `  F
) ) )  <->  ( ( R `  h )
( join `  K )
( R `  F
) ) ( le
`  K ) ( ( R `  (
h  o.  F ) ) ( join `  K
) ( R `  F ) ) ) )
5946, 48, 58mpbi2and 892 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( ( R `  h )
( join `  K )
( R `  F
) ) ( le
`  K ) ( ( R `  (
h  o.  F ) ) ( join `  K
) ( R `  F ) ) )
6024, 25, 62atjlej 28798 . . 3  |-  ( ( K  e.  HL  /\  ( ( R `  h )  e.  (
Atoms `  K )  /\  ( R `  F )  e.  ( Atoms `  K
)  /\  ( R `  h )  =/=  ( R `  F )
)  /\  ( ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )  /\  ( R `  F
)  e.  ( Atoms `  K )  /\  (
( R `  h
) ( join `  K
) ( R `  F ) ) ( le `  K ) ( ( R `  ( h  o.  F
) ) ( join `  K ) ( R `
 F ) ) ) )  ->  ( R `  ( h  o.  F ) )  =/=  ( R `  F
) )
611, 12, 17, 18, 19, 17, 59, 60syl133anc 1210 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K )
)  ->  ( R `  ( h  o.  F
) )  =/=  ( R `  F )
)
62 nelne2 2509 . . . 4  |-  ( ( ( R `  F
)  e.  ( Atoms `  K )  /\  -.  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K
) )  ->  ( R `  F )  =/=  ( R `  (
h  o.  F ) ) )
6362necomd 2502 . . 3  |-  ( ( ( R `  F
)  e.  ( Atoms `  K )  /\  -.  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K
) )  ->  ( R `  ( h  o.  F ) )  =/=  ( R `  F
) )
6416, 63sylan 459 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F )
) )  /\  -.  ( R `  ( h  o.  F ) )  e.  ( Atoms `  K
) )  ->  ( R `  ( h  o.  F ) )  =/=  ( R `  F
) )
6561, 64pm2.61dan 769 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  h  e.  T )  /\  ( F  =/=  (  _I  |`  B )  /\  h  =/=  (  _I  |`  B )  /\  ( R `  h )  =/=  ( R `  F ) ) )  ->  ( R `  ( h  o.  F
) )  =/=  ( R `  F )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963    _I cid 4241   `'ccnv 4625    |` cres 4628    o. ccom 4630   -->wf 4634   -1-1-onto->wf1o 4637   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   Latclat 14078   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477
This theorem is referenced by:  cdlemg47  30055
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-llines 28817  df-lplanes 28818  df-lvols 28819  df-lines 28820  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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