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Theorem cdlemg31d 33975
Description: Eliminate  ( F `
 P )  =/= 
P from cdlemg31c 33974. 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 33980 vs. cdlemg28 33979. (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 1125 . . . 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 466 . . 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 1008 . . . . . . 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 1122 . . . . . . . 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 466 . . . . . . 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 1124 . . . . . . . 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 466 . . . . . . 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 1126 . . . . . . . 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 466 . . . . . . 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 1086 . . . . . . 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 33973 . . . . . . 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 1273 . . . . . 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 1083 . . . . . . . . 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 462 . . . . . . . . 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 2429 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
2411, 23, 14, 15, 16, 17trl0 33444 . . . . . . . . 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 1268 . . . . . . . 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 6321 . . . . . . 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 1029 . . . . . . . . . 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 32635 . . . . . . . . . 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 466 . . . . . . . 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 2429 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
3231, 14atbase 32563 . . . . . . . . 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 32499 . . . . . . . 8  |-  ( ( K  e.  OL  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  ( 0. `  K ) )  =  Q )
3530, 33, 34syl2anc 665 . . . . . . 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 2470 . . . . . 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 4450 . . . . 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 32634 . . . . . . . 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 466 . . . . . 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 1088 . . . . . 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 32585 . . . . . 6  |-  ( ( K  e.  AtLat  /\  N  e.  A  /\  Q  e.  A )  ->  ( N  .<_  Q  <->  N  =  Q ) )
4340, 41, 7, 42syl3anc 1264 . . . . 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 213 . . . 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 4436 . . 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 302 . 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 1008 . . 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 1083 . . 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 1084 . . 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 1085 . . 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 1086 . . 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 1087 . . 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 462 . . 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 1088 . . 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 33974 . . 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 1294 . 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 2749 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   0.cp0 16234   OLcol 32448   Atomscatm 32537   AtLatcal 32538   HLchlt 32624   LHypclh 33257   LTrncltrn 33374   trLctrl 33432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32450  df-ol 32452  df-oml 32453  df-covers 32540  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625  df-psubsp 32776  df-pmap 32777  df-padd 33069  df-lhyp 33261  df-laut 33262  df-ldil 33377  df-ltrn 33378  df-trl 33433
This theorem is referenced by:  cdlemg33b0  33976  cdlemg33a  33981
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