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Theorem cdlemg35 34676
Description: TODO: Fix comment. TODO: should we have a more general version of hlsupr 33349 to avoid the  =/= conditions? (Contributed by NM, 31-May-2013.)
Hypotheses
Ref Expression
cdlemg35.l  |-  .<_  =  ( le `  K )
cdlemg35.j  |-  .\/  =  ( join `  K )
cdlemg35.m  |-  ./\  =  ( meet `  K )
cdlemg35.a  |-  A  =  ( Atoms `  K )
cdlemg35.h  |-  H  =  ( LHyp `  K
)
cdlemg35.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg35.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg35  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  E. v  e.  A  ( v  .<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
) ) )
Distinct variable groups:    v, A    v, F    v, G    v, H    v, K    v,  .<_    v, P    v, R    v, T    v, W
Allowed substitution hints:    .\/ ( v)    ./\ ( v)

Proof of Theorem cdlemg35
StepHypRef Expression
1 simp1l 1012 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  K  e.  HL )
2 simp1 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp21 1021 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 simp22 1022 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  F  e.  T
)
5 simp31 1024 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( F `  P )  =/=  P
)
6 cdlemg35.l . . . . 5  |-  .<_  =  ( le `  K )
7 cdlemg35.a . . . . 5  |-  A  =  ( Atoms `  K )
8 cdlemg35.h . . . . 5  |-  H  =  ( LHyp `  K
)
9 cdlemg35.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
10 cdlemg35.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
116, 7, 8, 9, 10trlat 34132 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
122, 3, 4, 5, 11syl112anc 1223 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( R `  F )  e.  A
)
13 simp23 1023 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  G  e.  T
)
14 simp32 1025 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( G `  P )  =/=  P
)
156, 7, 8, 9, 10trlat 34132 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G  e.  T  /\  ( G `  P )  =/=  P ) )  ->  ( R `  G )  e.  A
)
162, 3, 13, 14, 15syl112anc 1223 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( R `  G )  e.  A
)
17 simp33 1026 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( R `  F )  =/=  ( R `  G )
)
18 cdlemg35.j . . . 4  |-  .\/  =  ( join `  K )
196, 18, 7hlsupr 33349 . . 3  |-  ( ( ( K  e.  HL  /\  ( R `  F
)  e.  A  /\  ( R `  G )  e.  A )  /\  ( R `  F )  =/=  ( R `  G ) )  ->  E. v  e.  A  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) ) )
201, 12, 16, 17, 19syl31anc 1222 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  E. v  e.  A  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) ) )
21 eqid 2452 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
22 simp11l 1099 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  ->  K  e.  HL )
23 hllat 33327 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
2422, 23syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  ->  K  e.  Lat )
2521, 7atbase 33253 . . . . . . 7  |-  ( v  e.  A  ->  v  e.  ( Base `  K
) )
26253ad2ant2 1010 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
v  e.  ( Base `  K ) )
27 simp11 1018 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
28 simp122 1121 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  ->  F  e.  T )
2921, 8, 9, 10trlcl 34127 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
3027, 28, 29syl2anc 661 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( R `  F
)  e.  ( Base `  K ) )
31 simp123 1122 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  ->  G  e.  T )
3221, 8, 9, 10trlcl 34127 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
3327, 31, 32syl2anc 661 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( R `  G
)  e.  ( Base `  K ) )
3421, 18latjcl 15335 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( R `  F )  e.  ( Base `  K
)  /\  ( R `  G )  e.  (
Base `  K )
)  ->  ( ( R `  F )  .\/  ( R `  G
) )  e.  (
Base `  K )
)
3524, 30, 33, 34syl3anc 1219 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( ( R `  F )  .\/  ( R `  G )
)  e.  ( Base `  K ) )
36 simp11r 1100 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  ->  W  e.  H )
3721, 8lhpbase 33961 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3836, 37syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  ->  W  e.  ( Base `  K ) )
39 simp33 1026 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
v  .<_  ( ( R `
 F )  .\/  ( R `  G ) ) )
406, 8, 9, 10trlle 34147 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
4127, 28, 40syl2anc 661 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( R `  F
)  .<_  W )
426, 8, 9, 10trlle 34147 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  .<_  W )
4327, 31, 42syl2anc 661 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( R `  G
)  .<_  W )
4421, 6, 18latjle12 15346 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( R `  F )  e.  (
Base `  K )  /\  ( R `  G
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( ( R `
 F )  .<_  W  /\  ( R `  G )  .<_  W )  <-> 
( ( R `  F )  .\/  ( R `  G )
)  .<_  W ) )
4524, 30, 33, 38, 44syl13anc 1221 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( ( ( R `
 F )  .<_  W  /\  ( R `  G )  .<_  W )  <-> 
( ( R `  F )  .\/  ( R `  G )
)  .<_  W ) )
4641, 43, 45mpbi2and 912 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( ( R `  F )  .\/  ( R `  G )
)  .<_  W )
4721, 6, 24, 26, 35, 38, 39, 46lattrd 15342 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
v  .<_  W )
48 simp31 1024 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
v  =/=  ( R `
 F ) )
49 simp32 1025 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
v  =/=  ( R `
 G ) )
5047, 48, 49jca32 535 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A  /\  (
v  =/=  ( R `
 F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F )  .\/  ( R `  G )
) ) )  -> 
( v  .<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
) ) )
51503expia 1190 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T
)  /\  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P  /\  ( R `  F )  =/=  ( R `  G )
) )  /\  v  e.  A )  ->  (
( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) )  ->  ( v  .<_  W  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) ) )
5251reximdva 2928 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  ( E. v  e.  A  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  v  .<_  ( ( R `  F ) 
.\/  ( R `  G ) ) )  ->  E. v  e.  A  ( v  .<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
) ) ) )
5320, 52mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  F  e.  T  /\  G  e.  T )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P  /\  ( R `  F )  =/=  ( R `  G ) ) )  ->  E. v  e.  A  ( v  .<_  W  /\  ( v  =/=  ( R `  F )  /\  v  =/=  ( R `  G )
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   E.wrex 2797   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   Latclat 15329   Atomscatm 33227   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   trLctrl 34121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122
This theorem is referenced by:  cdlemg36  34677
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