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Theorem cdlemg31d 34312
Description: Eliminate  ( F `
 P )  =/= 
P from cdlemg31c 34311. TODO: Prove directly. TODO: do we need to eliminate  ( F `  P )  =/=  P? It might be better to do this all at once at the end. See also cdlemg29 34317 vs. cdlemg28 34316. (Contributed by NM, 29-May-2013.)
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 )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
Assertion
Ref Expression
cdlemg31d  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  -.  N  .<_  W )

Proof of Theorem cdlemg31d
StepHypRef Expression
1 simp22r 1134 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  -.  Q  .<_  W )
21adantr 471 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  -.  Q  .<_  W )
3 simpl1 1017 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 simp21l 1131 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  P  e.  A )
54adantr 471 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  P  e.  A )
6 simp22l 1133 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  Q  e.  A )
76adantr 471 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  Q  e.  A )
8 simp23l 1135 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  v  e.  A )
98adantr 471 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
v  e.  A )
10 simpl31 1095 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  F  e.  T )
11 cdlemg12.l . . . . . . . 8  |-  .<_  =  ( le `  K )
12 cdlemg12.j . . . . . . . 8  |-  .\/  =  ( join `  K )
13 cdlemg12.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
14 cdlemg12.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
15 cdlemg12.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
16 cdlemg12.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
17 cdlemg12b.r . . . . . . . 8  |-  R  =  ( ( trL `  K
) `  W )
18 cdlemg31.n . . . . . . . 8  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
1911, 12, 13, 14, 15, 16, 17, 18cdlemg31b 34310 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  N  .<_  ( Q  .\/  ( R `
 F ) ) )
203, 5, 7, 9, 10, 19syl122anc 1285 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  N  .<_  ( Q  .\/  ( R `  F ) ) )
21 simpl21 1092 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
22 simpr 467 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( F `  P
)  =  P )
23 eqid 2462 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
2411, 23, 14, 15, 16, 17trl0 33781 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( 0. `  K ) )
253, 21, 10, 22, 24syl112anc 1280 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( R `  F
)  =  ( 0.
`  K ) )
2625oveq2d 6331 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( Q  .\/  ( R `  F )
)  =  ( Q 
.\/  ( 0. `  K ) ) )
27 simp1l 1038 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  K  e.  HL )
28 hlol 32972 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
2927, 28syl 17 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  K  e.  OL )
3029adantr 471 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  K  e.  OL )
31 eqid 2462 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
3231, 14atbase 32900 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
337, 32syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  Q  e.  ( Base `  K ) )
3431, 12, 23olj01 32836 . . . . . . . 8  |-  ( ( K  e.  OL  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  ( 0. `  K ) )  =  Q )
3530, 33, 34syl2anc 671 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( Q  .\/  ( 0. `  K ) )  =  Q )
3626, 35eqtrd 2496 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( Q  .\/  ( R `  F )
)  =  Q )
3720, 36breqtrd 4441 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  N  .<_  Q )
38 hlatl 32971 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
3927, 38syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  K  e.  AtLat )
4039adantr 471 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  K  e.  AtLat )
41 simpl33 1097 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  N  e.  A )
4211, 14atcmp 32922 . . . . . 6  |-  ( ( K  e.  AtLat  /\  N  e.  A  /\  Q  e.  A )  ->  ( N  .<_  Q  <->  N  =  Q ) )
4340, 41, 7, 42syl3anc 1276 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( N  .<_  Q  <->  N  =  Q ) )
4437, 43mpbid 215 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  N  =  Q )
4544breq1d 4426 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  -> 
( N  .<_  W  <->  Q  .<_  W ) )
462, 45mtbird 307 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =  P )  ->  -.  N  .<_  W )
47 simpl1 1017 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  -> 
( K  e.  HL  /\  W  e.  H ) )
48 simpl21 1092 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
49 simpl22 1093 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
50 simpl23 1094 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  -> 
( v  e.  A  /\  v  .<_  W ) )
51 simpl31 1095 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  ->  F  e.  T )
52 simpl32 1096 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  -> 
v  =/=  ( R `
 F ) )
53 simpr 467 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  -> 
( F `  P
)  =/=  P )
54 simpl33 1097 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  ->  N  e.  A )
5511, 12, 13, 14, 15, 16, 17, 18cdlemg31c 34311 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  -.  N  .<_  W )
5647, 48, 49, 50, 51, 52, 53, 54, 55syl323anc 1306 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F
)  /\  N  e.  A ) )  /\  ( F `  P )  =/=  P )  ->  -.  N  .<_  W )
5746, 56pm2.61dane 2723 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  -.  N  .<_  W )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   Basecbs 15170   lecple 15246   joincjn 16238   meetcmee 16239   0.cp0 16332   OLcol 32785   Atomscatm 32874   AtLatcal 32875   HLchlt 32961   LHypclh 33594   LTrncltrn 33711   trLctrl 33769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-map 7500  df-preset 16222  df-poset 16240  df-plt 16253  df-lub 16269  df-glb 16270  df-join 16271  df-meet 16272  df-p0 16334  df-p1 16335  df-lat 16341  df-clat 16403  df-oposet 32787  df-ol 32789  df-oml 32790  df-covers 32877  df-ats 32878  df-atl 32909  df-cvlat 32933  df-hlat 32962  df-psubsp 33113  df-pmap 33114  df-padd 33406  df-lhyp 33598  df-laut 33599  df-ldil 33714  df-ltrn 33715  df-trl 33770
This theorem is referenced by:  cdlemg33b0  34313  cdlemg33a  34318
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