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Theorem cdleme1 30709
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 731 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  HL )
2 simpr3l 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  A )
3 hllat 29846 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
43ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  Lat )
5 eqid 2404 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atbase 29772 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
82, 7syl 16 . . . . 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 963 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  A )
115, 6atbase 29772 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1210, 11syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  ( Base `  K )
)
13 simpr2 964 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  A )
145, 6atbase 29772 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1513, 14syl 16 . . . . . . . 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 14434 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
184, 12, 15, 17syl3anc 1184 . . . . . . 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 30480 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2120ad2antlr 708 . . . . . . 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 14435 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
244, 18, 21, 23syl3anc 1184 . . . . . 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 2488 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  e.  ( Base `  K )
)
265, 16latjcl 14434 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( R  .\/  U )  e.  ( Base `  K
) )
274, 8, 25, 26syl3anc 1184 . . . 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 14434 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
294, 12, 8, 28syl3anc 1184 . . . . . 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 14435 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R )  ./\  W )  e.  ( Base `  K ) )
314, 29, 21, 30syl3anc 1184 . . . . 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 14434 . . . . 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 1184 . . . 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 14444 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  R  .<_  ( R  .\/  U
) )
364, 8, 25, 35syl3anc 1184 . . . 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 30346 . . . 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 1197 . . 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 14445 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  R  .<_  ( P  .\/  R
) )
404, 12, 8, 39syl3anc 1184 . . . . . . . 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 30346 . . . . . . . 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 1197 . . . . . . 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 2404 . . . . . . . . . 10  |-  ( 1.
`  K )  =  ( 1. `  K
)
4434, 16, 43, 6, 19lhpjat2 30503 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  .\/  W
)  =  ( 1.
`  K ) )
45443ad2antr3 1124 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  W )  =  ( 1. `  K ) )
4645oveq2d 6056 . . . . . . 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 29844 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OL )
4847ad2antrr 707 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  OL )
495, 22, 43olm11 29710 . . . . . . . 8  |-  ( ( K  e.  OL  /\  ( P  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  R
)  ./\  ( 1. `  K ) )  =  ( P  .\/  R
) )
5048, 29, 49syl2anc 643 . . . . . . 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 2440 . . . . . 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 6056 . . . . 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 14480 . . . . . 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 1186 . . . . 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 14482 . . . . . 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 1186 . . . . 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 2445 . . . 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 6056 . . 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 30702 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
60593adantr3 1118 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  .<_  ( P  .\/  Q ) )
615, 34, 16latjlej2 14450 . . . . . 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 1186 . . . . 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 15 . . . 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 14434 . . . . . 6  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( R  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)
654, 8, 18, 64syl3anc 1184 . . . . 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 14463 . . . . 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 1184 . . . 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 202 . . 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 2443 . 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 6051 . 2  |-  ( R 
.\/  F )  =  ( R  .\/  (
( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
7269, 71syl6reqr 2455 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 3    -> wi 4    <-> wb 177    /\ 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
This theorem is referenced by:  cdleme2  30710  cdleme3b  30711  cdleme3c  30712  cdleme5  30722  cdleme11  30752  cdleme12  30753  cdleme16c  30762  cdleme20g  30797  cdleme35a  30930  cdleme36a  30942
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-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
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