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Theorem cdlemg35 34351
Description: TODO: Fix comment. TODO: should we have a more general version of hlsupr 33022 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 1054 . . 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 1030 . . . 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 1063 . . . 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 1064 . . . 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 1066 . . . 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 33806 . . . 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 1296 . . 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 1065 . . . 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 1067 . . . 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 33806 . . . 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 1296 . . 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 1068 . . 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 33022 . . 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 1295 . 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 2471 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
22 simp11l 1141 . . . . . . 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 33000 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
2422, 23syl 17 . . . . . 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 32926 . . . . . . 7  |-  ( v  e.  A  ->  v  e.  ( Base `  K
) )
26253ad2ant2 1052 . . . . . 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 1060 . . . . . . . 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 1163 . . . . . . . 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 33801 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
3027, 28, 29syl2anc 673 . . . . . . 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 1164 . . . . . . . 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 33801 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
3327, 31, 32syl2anc 673 . . . . . . 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 16375 . . . . . . 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 1292 . . . . . 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 1142 . . . . . . 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 33634 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3836, 37syl 17 . . . . . 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 1068 . . . . . 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 33821 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
4127, 28, 40syl2anc 673 . . . . . . 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 33821 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  .<_  W )
4327, 31, 42syl2anc 673 . . . . . . 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 16386 . . . . . . . 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 1294 . . . . . . 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 935 . . . . . 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 16382 . . . . 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 1066 . . . . 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 1067 . . . . 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 544 . . . 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 1233 . . 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 2858 . 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   lecple 15275   joincjn 16267   meetcmee 16268   Latclat 16369   Atomscatm 32900   HLchlt 32987   LHypclh 33620   LTrncltrn 33737   trLctrl 33795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796
This theorem is referenced by:  cdlemg36  34352
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