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Theorem cdleme1 29666
Description: Part of proof of Lemma E in [Crawley] p. 113.  F represents their f(r). Here we show r  \/ f(r) = r  \/ u (7th through 5th lines from bottom on p. 113). (Contributed by NM, 4-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l  |-  .<_  =  ( le `  K )
cdleme1.j  |-  .\/  =  ( join `  K )
cdleme1.m  |-  ./\  =  ( meet `  K )
cdleme1.a  |-  A  =  ( Atoms `  K )
cdleme1.h  |-  H  =  ( LHyp `  K
)
cdleme1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme1.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
Assertion
Ref Expression
cdleme1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  =  ( R  .\/  U ) )

Proof of Theorem cdleme1
StepHypRef Expression
1 simpll 733 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  HL )
2 simpr3l 1021 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  A )
3 hllat 28803 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
43ad2antrr 709 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  Lat )
5 eqid 2258 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atbase 28729 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
82, 7syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  ( Base `  K )
)
9 cdleme1.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
10 simpr1 966 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  A )
115, 6atbase 28729 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1210, 11syl 17 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  ( Base `  K )
)
13 simpr2 967 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  A )
145, 6atbase 28729 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1513, 14syl 17 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  ( Base `  K )
)
16 cdleme1.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
175, 16latjcl 14119 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
184, 12, 15, 17syl3anc 1187 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
19 cdleme1.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
205, 19lhpbase 29437 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2120ad2antlr 710 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  W  e.  ( Base `  K )
)
22 cdleme1.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
235, 22latmcl 14120 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
244, 18, 21, 23syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
259, 24syl5eqel 2342 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  e.  ( Base `  K )
)
265, 16latjcl 14119 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( R  .\/  U )  e.  ( Base `  K
) )
274, 8, 25, 26syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  U )  e.  (
Base `  K )
)
285, 16latjcl 14119 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
294, 12, 8, 28syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( P  .\/  R )  e.  (
Base `  K )
)
305, 22latmcl 14120 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R )  ./\  W )  e.  ( Base `  K ) )
314, 29, 21, 30syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P  .\/  R )  ./\  W )  e.  ( Base `  K ) )
325, 16latjcl 14119 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  R
)  ./\  W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  (
Base `  K )
)
334, 15, 31, 32syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
)  e.  ( Base `  K ) )
34 cdleme1.l . . . . . 6  |-  .<_  =  ( le `  K )
355, 34, 16latlej1 14129 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  R  .<_  ( R  .\/  U
) )
364, 8, 25, 35syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  .<_  ( R  .\/  U ) )
375, 34, 16, 22, 6atmod3i1 29303 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  ( R  .\/  U
)  e.  ( Base `  K )  /\  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  (
Base `  K )
)  /\  R  .<_  ( R  .\/  U ) )  ->  ( R  .\/  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )  =  ( ( R  .\/  U )  ./\  ( R  .\/  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
) ) ) )
381, 2, 27, 33, 36, 37syl131anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )  =  ( ( R  .\/  U )  ./\  ( R  .\/  ( Q  .\/  (
( P  .\/  R
)  ./\  W )
) ) ) )
395, 34, 16latlej2 14130 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  R  .<_  ( P  .\/  R
) )
404, 12, 8, 39syl3anc 1187 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  .<_  ( P  .\/  R ) )
415, 34, 16, 22, 6atmod3i1 29303 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  ( P  .\/  R
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  R  .<_  ( P  .\/  R
) )  ->  ( R  .\/  ( ( P 
.\/  R )  ./\  W ) )  =  ( ( P  .\/  R
)  ./\  ( R  .\/  W ) ) )
421, 2, 29, 21, 40, 41syl131anc 1200 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  ( ( P  .\/  R )  ./\  W )
)  =  ( ( P  .\/  R ) 
./\  ( R  .\/  W ) ) )
43 eqid 2258 . . . . . . . . . 10  |-  ( 1.
`  K )  =  ( 1. `  K
)
4434, 16, 43, 6, 19lhpjat2 29460 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  .\/  W
)  =  ( 1.
`  K ) )
45443ad2antr3 1127 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  W )  =  ( 1. `  K ) )
4645oveq2d 5808 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P  .\/  R )  ./\  ( R  .\/  W ) )  =  ( ( P  .\/  R ) 
./\  ( 1. `  K ) ) )
47 hlol 28801 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OL )
4847ad2antrr 709 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  OL )
495, 22, 43olm11 28667 . . . . . . . 8  |-  ( ( K  e.  OL  /\  ( P  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  R
)  ./\  ( 1. `  K ) )  =  ( P  .\/  R
) )
5048, 29, 49syl2anc 645 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( P  .\/  R )  ./\  ( 1. `  K ) )  =  ( P 
.\/  R ) )
5142, 46, 503eqtrd 2294 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  ( ( P  .\/  R )  ./\  W )
)  =  ( P 
.\/  R ) )
5251oveq2d 5808 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( Q  .\/  ( R  .\/  (
( P  .\/  R
)  ./\  W )
) )  =  ( Q  .\/  ( P 
.\/  R ) ) )
535, 16latj12 14165 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  (
( P  .\/  R
)  ./\  W )  e.  ( Base `  K
) ) )  -> 
( Q  .\/  ( R  .\/  ( ( P 
.\/  R )  ./\  W ) ) )  =  ( R  .\/  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) ) ) )
544, 15, 8, 31, 53syl13anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( Q  .\/  ( R  .\/  (
( P  .\/  R
)  ./\  W )
) )  =  ( R  .\/  ( Q 
.\/  ( ( P 
.\/  R )  ./\  W ) ) ) )
555, 16latj13 14167 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) ) )  -> 
( Q  .\/  ( P  .\/  R ) )  =  ( R  .\/  ( P  .\/  Q ) ) )
564, 15, 12, 8, 55syl13anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( Q  .\/  ( P  .\/  R
) )  =  ( R  .\/  ( P 
.\/  Q ) ) )
5752, 54, 563eqtr3rd 2299 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  ( P  .\/  Q
) )  =  ( R  .\/  ( Q 
.\/  ( ( P 
.\/  R )  ./\  W ) ) ) )
5857oveq2d 5808 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  .\/  U )  ./\  ( R  .\/  ( P 
.\/  Q ) ) )  =  ( ( R  .\/  U ) 
./\  ( R  .\/  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) ) ) )
5934, 16, 22, 6, 19, 9cdlemeulpq 29659 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
60593adantr3 1121 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  .<_  ( P  .\/  Q ) )
615, 34, 16latjlej2 14135 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
) )  ->  ( U  .<_  ( P  .\/  Q )  ->  ( R  .\/  U )  .<_  ( R 
.\/  ( P  .\/  Q ) ) ) )
624, 25, 18, 8, 61syl13anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( U  .<_  ( P  .\/  Q
)  ->  ( R  .\/  U )  .<_  ( R 
.\/  ( P  .\/  Q ) ) ) )
6360, 62mpd 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  U )  .<_  ( R 
.\/  ( P  .\/  Q ) ) )
645, 16latjcl 14119 . . . . . 6  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( R  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)
654, 8, 18, 64syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  ( P  .\/  Q
) )  e.  (
Base `  K )
)
665, 34, 22latleeqm1 14148 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  U )  e.  ( Base `  K
)  /\  ( R  .\/  ( P  .\/  Q
) )  e.  (
Base `  K )
)  ->  ( ( R  .\/  U )  .<_  ( R  .\/  ( P 
.\/  Q ) )  <-> 
( ( R  .\/  U )  ./\  ( R  .\/  ( P  .\/  Q
) ) )  =  ( R  .\/  U
) ) )
674, 27, 65, 66syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  .\/  U )  .<_  ( R  .\/  ( P 
.\/  Q ) )  <-> 
( ( R  .\/  U )  ./\  ( R  .\/  ( P  .\/  Q
) ) )  =  ( R  .\/  U
) ) )
6863, 67mpbid 203 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( ( R  .\/  U )  ./\  ( R  .\/  ( P 
.\/  Q ) ) )  =  ( R 
.\/  U ) )
6938, 58, 683eqtr2rd 2297 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  U )  =  ( R  .\/  ( ( R  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) ) )
70 cdleme1.f . . 3  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
7170oveq2i 5803 . 2  |-  ( R 
.\/  F )  =  ( R  .\/  (
( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
7269, 71syl6reqr 2309 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  F )  =  ( R  .\/  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   1.cp1 14107   Latclat 14114   OLcol 28614   Atomscatm 28703   HLchlt 28790   LHypclh 29423
This theorem is referenced by:  cdleme2  29667  cdleme3b  29668  cdleme3c  29669  cdleme5  29679  cdleme11  29709  cdleme12  29710  cdleme16c  29719  cdleme20g  29754  cdleme35a  29887  cdleme36a  29899
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427
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