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Theorem cdlemn10 31689
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 981 . . 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 29846 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . 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 1076 . . 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 29772 . . 3  |-  ( S  e.  A  ->  S  e.  B )
96, 8syl 16 . 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 1074 . . 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 29849 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  e.  B )
133, 10, 6, 12syl3anc 1184 . 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 29772 . . . 4  |-  ( Q  e.  A  ->  Q  e.  B )
1510, 14syl 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  e.  B )
16 simp23l 1078 . . 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 14434 . . 3  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  X  e.  B )  ->  ( Q  .\/  X
)  e.  B )
185, 15, 16, 17syl3anc 1184 . 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 29858 . . 3  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  S  .<_  ( Q  .\/  S ) )
203, 10, 6, 19syl3anc 1184 . 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 982 . . . . . 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 30480 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2421, 23syl 16 . . . . 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 29857 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  Q  .<_  ( Q  .\/  S ) )
263, 10, 6, 25syl3anc 1184 . . . . 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 2404 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
281, 2, 11, 27, 7atmod3i1 30346 . . . . 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 1197 . . . 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 957 . . . . . 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 990 . . . . . 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 2404 . . . . . . 7  |-  ( 1.
`  K )  =  ( 1. `  K
)
332, 11, 32, 7, 22lhpjat2 30503 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  .\/  W
)  =  ( 1.
`  K ) )
3430, 31, 33syl2anc 643 . . . . 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 6056 . . . 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 29844 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
373, 36syl 16 . . . . 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 29710 . . . . 5  |-  ( ( K  e.  OL  /\  ( Q  .\/  S )  e.  B )  -> 
( ( Q  .\/  S ) ( meet `  K
) ( 1. `  K ) )  =  ( Q  .\/  S
) )
3937, 13, 38syl2anc 643 . . . 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 2441 . . 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 993 . . . . . . 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 30645 . . . . . . 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 1184 . . . . . 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 994 . . . . . . . 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 6056 . . . . . . 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 6055 . . . . . 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 2436 . . . . 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 995 . . . . 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 4194 . . . 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 14435 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( Q  .\/  S
) ( meet `  K
) W )  e.  B )
535, 13, 24, 52syl3anc 1184 . . . . 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 14450 . . . . 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 1186 . . . 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 15 . . 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 4192 . 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 14442 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 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   1.cp1 14422   Latclat 14429   OLcol 29657   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  cdlemn11pre  31693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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