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Theorem cdlemg16ALTN 34269
Description: This version of cdlemg16 34268 uses cdlemg15a 34266 instead of cdlemg15 34267, in case cdlemg15 34267 ends up not being needed. TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg16ALTN  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )

Proof of Theorem cdlemg16ALTN
StepHypRef Expression
1 simpl11 1089 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  ->  K  e.  HL )
2 simpl12 1090 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  ->  W  e.  H )
31, 2jca 539 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 simpl21 1092 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
5 simpl22 1093 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
6 simpl13 1091 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  -> 
( F  e.  T  /\  G  e.  T
) )
7 simpr 467 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  -> 
( R `  F
)  =  ( R `
 G ) )
8 simpl31 1095 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  -> 
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )
9 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
10 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
11 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
12 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
13 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
14 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
15 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
169, 10, 11, 12, 13, 14, 15cdlemg15a 34266 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( R `  F )  =  ( R `  G )  /\  (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
173, 4, 5, 6, 7, 8, 16syl312anc 1297 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =  ( R `  G ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
18 simpl11 1089 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  ->  K  e.  HL )
19 simpl12 1090 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  ->  W  e.  H )
2018, 19jca 539 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
21 simpl21 1092 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
22 simpl22 1093 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
23 simp13l 1129 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  F  e.  T )
2423adantr 471 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  ->  F  e.  T )
25 simp13r 1130 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  ->  G  e.  T )
2625adantr 471 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  ->  G  e.  T )
27 simpl23 1094 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  ->  P  =/=  Q )
28 simpl32 1096 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  ->  -.  ( R `  F
)  .<_  ( P  .\/  Q ) )
29 simpl33 1097 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  ->  -.  ( R `  G
)  .<_  ( P  .\/  Q ) )
3028, 29jca 539 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  -> 
( -.  ( R `
 F )  .<_  ( P  .\/  Q )  /\  -.  ( R `
 G )  .<_  ( P  .\/  Q ) ) )
31 simpr 467 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  -> 
( R `  F
)  =/=  ( R `
 G ) )
32 simpl31 1095 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  -> 
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )
339, 10, 11, 12, 13, 14, 15cdlemg12 34261 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( ( -.  ( R `  F
)  .<_  ( P  .\/  Q )  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) )  /\  ( R `  F )  =/=  ( R `  G
)  /\  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
3420, 21, 22, 24, 26, 27, 30, 31, 32, 33syl333anc 1308 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q
)  /\  ( (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
)  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( R `  F )  =/=  ( R `  G ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
3517, 34pm2.61dane 2722 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( F  e.  T  /\  G  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  P  =/=  Q )  /\  (
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  ( R `  F )  .<_  ( P  .\/  Q
)  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   lecple 15245   joincjn 16237   meetcmee 16238   Atomscatm 32873   HLchlt 32960   LHypclh 33593   LTrncltrn 33710   trLctrl 33768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-riotaBAD 32569
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-1st 6819  df-2nd 6820  df-undef 7045  df-map 7499  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-p1 16334  df-lat 16340  df-clat 16402  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108  df-lvols 33109  df-lines 33110  df-psubsp 33112  df-pmap 33113  df-padd 33405  df-lhyp 33597  df-laut 33598  df-ldil 33713  df-ltrn 33714  df-trl 33769
This theorem is referenced by: (None)
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