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Theorem cdlemn10 30526
Description: Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
Hypotheses
Ref Expression
cdlemn10.b  |-  B  =  ( Base `  K
)
cdlemn10.l  |-  .<_  =  ( le `  K )
cdlemn10.j  |-  .\/  =  ( join `  K )
cdlemn10.a  |-  A  =  ( Atoms `  K )
cdlemn10.h  |-  H  =  ( LHyp `  K
)
cdlemn10.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemn10.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemn10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  S  .<_  ( Q  .\/  X ) )

Proof of Theorem cdlemn10
StepHypRef Expression
1 cdlemn10.b . 2  |-  B  =  ( Base `  K
)
2 cdlemn10.l . 2  |-  .<_  =  ( le `  K )
3 simp1l 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  K  e.  HL )
4 hllat 28683 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 17 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  K  e.  Lat )
6 simp22l 1079 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  S  e.  A )
7 cdlemn10.a . . . 4  |-  A  =  ( Atoms `  K )
81, 7atbase 28609 . . 3  |-  ( S  e.  A  ->  S  e.  B )
96, 8syl 17 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  S  e.  B )
10 simp21l 1077 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  Q  e.  A )
11 cdlemn10.j . . . 4  |-  .\/  =  ( join `  K )
121, 11, 7hlatjcl 28686 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  B )
133, 10, 6, 12syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  .\/  S
)  e.  B )
141, 7atbase 28609 . . . 4  |-  ( Q  e.  A  ->  Q  e.  B )
1510, 14syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  Q  e.  B )
16 simp23l 1081 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  X  e.  B )
171, 11latjcl 14083 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  e.  B )
185, 15, 16, 17syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  .\/  X
)  e.  B )
192, 11, 7hlatlej2 28695 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  S  .<_  ( Q  .\/  S ) )
203, 10, 6, 19syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  S  .<_  ( Q  .\/  S ) )
21 simp1r 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  W  e.  H )
22 cdlemn10.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
231, 22lhpbase 29317 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2421, 23syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  W  e.  B )
252, 11, 7hlatlej1 28694 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  Q  .<_  ( Q  .\/  S ) )
263, 10, 6, 25syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  Q  .<_  ( Q  .\/  S ) )
27 eqid 2256 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
281, 2, 11, 27, 7atmod3i1 29183 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( Q  .\/  S
)  e.  B  /\  W  e.  B )  /\  Q  .<_  ( Q 
.\/  S ) )  ->  ( Q  .\/  ( ( Q  .\/  S ) ( meet `  K
) W ) )  =  ( ( Q 
.\/  S ) (
meet `  K )
( Q  .\/  W
) ) )
293, 10, 13, 24, 26, 28syl131anc 1200 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  .\/  (
( Q  .\/  S
) ( meet `  K
) W ) )  =  ( ( Q 
.\/  S ) (
meet `  K )
( Q  .\/  W
) ) )
30 simp1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
31 simp21 993 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
32 eqid 2256 . . . . . . 7  |-  ( 1.
`  K )  =  ( 1. `  K
)
332, 11, 32, 7, 22lhpjat2 29340 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  .\/  W
)  =  ( 1.
`  K ) )
3430, 31, 33syl2anc 645 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  .\/  W
)  =  ( 1.
`  K ) )
3534oveq2d 5773 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( ( Q  .\/  S ) ( meet `  K
) ( Q  .\/  W ) )  =  ( ( Q  .\/  S
) ( meet `  K
) ( 1. `  K ) ) )
36 hlol 28681 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
373, 36syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  K  e.  OL )
381, 27, 32olm11 28547 . . . . 5  |-  ( ( K  e.  OL  /\  ( Q  .\/  S )  e.  B )  -> 
( ( Q  .\/  S ) ( meet `  K
) ( 1. `  K ) )  =  ( Q  .\/  S
) )
3937, 13, 38syl2anc 645 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( ( Q  .\/  S ) ( meet `  K
) ( 1. `  K ) )  =  ( Q  .\/  S
) )
4029, 35, 393eqtrrd 2293 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  .\/  S
)  =  ( Q 
.\/  ( ( Q 
.\/  S ) (
meet `  K ) W ) ) )
41 simp31 996 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
g  e.  T )
42 cdlemn10.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
43 cdlemn10.r . . . . . . . 8  |-  R  =  ( ( trL `  K
) `  W )
442, 11, 27, 7, 22, 42, 43trlval2 29482 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  g  e.  T  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( R `  g )  =  ( ( Q  .\/  (
g `  Q )
) ( meet `  K
) W ) )
4530, 41, 31, 44syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( R `  g
)  =  ( ( Q  .\/  ( g `
 Q ) ) ( meet `  K
) W ) )
46 simp32 997 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( g `  Q
)  =  S )
4746oveq2d 5773 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  .\/  (
g `  Q )
)  =  ( Q 
.\/  S ) )
4847oveq1d 5772 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( ( Q  .\/  ( g `  Q
) ) ( meet `  K ) W )  =  ( ( Q 
.\/  S ) (
meet `  K ) W ) )
4945, 48eqtrd 2288 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( R `  g
)  =  ( ( Q  .\/  S ) ( meet `  K
) W ) )
50 simp33 998 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( R `  g
)  .<_  X )
5149, 50eqbrtrrd 3985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( ( Q  .\/  S ) ( meet `  K
) W )  .<_  X )
521, 27latmcl 14084 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( Q  .\/  S
) ( meet `  K
) W )  e.  B )
535, 13, 24, 52syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( ( Q  .\/  S ) ( meet `  K
) W )  e.  B )
541, 2, 11latjlej2 14099 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( Q 
.\/  S ) (
meet `  K ) W )  e.  B  /\  X  e.  B  /\  Q  e.  B
) )  ->  (
( ( Q  .\/  S ) ( meet `  K
) W )  .<_  X  ->  ( Q  .\/  ( ( Q  .\/  S ) ( meet `  K
) W ) ) 
.<_  ( Q  .\/  X
) ) )
555, 53, 16, 15, 54syl13anc 1189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( ( ( Q 
.\/  S ) (
meet `  K ) W )  .<_  X  -> 
( Q  .\/  (
( Q  .\/  S
) ( meet `  K
) W ) ) 
.<_  ( Q  .\/  X
) ) )
5651, 55mpd 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  .\/  (
( Q  .\/  S
) ( meet `  K
) W ) ) 
.<_  ( Q  .\/  X
) )
5740, 56eqbrtrd 3983 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  -> 
( Q  .\/  S
)  .<_  ( Q  .\/  X ) )
581, 2, 5, 9, 13, 18, 20, 57lattrd 14091 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  (
g  e.  T  /\  ( g `  Q
)  =  S  /\  ( R `  g ) 
.<_  X ) )  ->  S  .<_  ( Q  .\/  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   1.cp1 14071   Latclat 14078   OLcol 28494   Atomscatm 28583   HLchlt 28670   LHypclh 29303   LTrncltrn 29420   trLctrl 29477
This theorem is referenced by:  cdlemn11pre  30530
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-map 6707  df-poset 14007  df-plt 14019  df-lub 14035  df-glb 14036  df-join 14037  df-meet 14038  df-p0 14072  df-p1 14073  df-lat 14079  df-clat 14141  df-oposet 28496  df-ol 28498  df-oml 28499  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-psubsp 28822  df-pmap 28823  df-padd 29115  df-lhyp 29307  df-laut 29308  df-ldil 29423  df-ltrn 29424  df-trl 29478
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