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Theorem cdlemg16ALTN 34137
Description: This version of cdlemg16 34136 uses cdlemg15a 34134 instead of cdlemg15 34135, in case cdlemg15 34135 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 1080 . . . 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 1081 . . . 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 534 . . 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 1083 . . 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 1084 . . 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 1082 . . 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 462 . . 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 1086 . . 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 34134 . . 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 1285 . 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 1080 . . . 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 1081 . . . 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 534 . . 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 1083 . . 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 1084 . . 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 1120 . . . 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 466 . . 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 1121 . . . 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 466 . . 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 1085 . . 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 1087 . . . 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 1088 . . . 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 534 . . 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 462 . . 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 1086 . . 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 34129 . . 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 1296 . 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 2683 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2594   class class class wbr 4361   ` cfv 5539  (class class class)co 6244   lecple 15135   joincjn 16127   meetcmee 16128   Atomscatm 32741   HLchlt 32828   LHypclh 33461   LTrncltrn 33578   trLctrl 33636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-riotaBAD 32437
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-1st 6746  df-2nd 6747  df-undef 6970  df-map 7424  df-preset 16111  df-poset 16129  df-plt 16142  df-lub 16158  df-glb 16159  df-join 16160  df-meet 16161  df-p0 16223  df-p1 16224  df-lat 16230  df-clat 16292  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-llines 32975  df-lplanes 32976  df-lvols 32977  df-lines 32978  df-psubsp 32980  df-pmap 32981  df-padd 33273  df-lhyp 33465  df-laut 33466  df-ldil 33581  df-ltrn 33582  df-trl 33637
This theorem is referenced by: (None)
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