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Theorem cdleme1 33226
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 752 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  HL )
2 simpr3l 1058 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  R  e.  A )
3 hllat 32362 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
43ad2antrr 724 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  Lat )
5 eqid 2402 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
6 cdleme1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atbase 32288 . . . . . 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 1003 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  P  e.  A )
115, 6atbase 32288 . . . . . . . . 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 1004 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  Q  e.  A )
145, 6atbase 32288 . . . . . . . . 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 15897 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
184, 12, 15, 17syl3anc 1230 . . . . . . 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 32996 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2120ad2antlr 725 . . . . . . 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 15898 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
244, 18, 21, 23syl3anc 1230 . . . . . 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 2494 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  e.  ( Base `  K )
)
265, 16latjcl 15897 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( R  .\/  U )  e.  ( Base `  K
) )
274, 8, 25, 26syl3anc 1230 . . . 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 15897 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
294, 12, 8, 28syl3anc 1230 . . . . . 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 15898 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R )  ./\  W )  e.  ( Base `  K ) )
314, 29, 21, 30syl3anc 1230 . . . . 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 15897 . . . . 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 1230 . . . 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 15906 . . . . 5  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  R  .<_  ( R  .\/  U
) )
364, 8, 25, 35syl3anc 1230 . . . 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 32862 . . . 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 1243 . . 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 15907 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  R  .<_  ( P  .\/  R
) )
404, 12, 8, 39syl3anc 1230 . . . . . . . 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 32862 . . . . . . . 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 1243 . . . . . . 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 2402 . . . . . . . . . 10  |-  ( 1.
`  K )  =  ( 1. `  K
)
4434, 16, 43, 6, 19lhpjat2 33019 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( R  .\/  W
)  =  ( 1.
`  K ) )
45443ad2antr3 1164 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( R  .\/  W )  =  ( 1. `  K ) )
4645oveq2d 6250 . . . . . . 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 32360 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OL )
4847ad2antrr 724 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  K  e.  OL )
495, 22, 43olm11 32226 . . . . . . . 8  |-  ( ( K  e.  OL  /\  ( P  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  R
)  ./\  ( 1. `  K ) )  =  ( P  .\/  R
) )
5048, 29, 49syl2anc 659 . . . . . . 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 2447 . . . . . 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 6250 . . . . 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 15942 . . . . . 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 1232 . . . . 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 15944 . . . . . 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 1232 . . . . 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 2452 . . . 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 6250 . . 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 33219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
60593adantr3 1158 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  U  .<_  ( P  .\/  Q ) )
615, 34, 16latjlej2 15912 . . . . . 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 1232 . . . . 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 15897 . . . . . 6  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( R  .\/  ( P  .\/  Q ) )  e.  (
Base `  K )
)
654, 8, 18, 64syl3anc 1230 . . . . 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 15925 . . . . 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 1230 . . . 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 210 . . 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 2450 . 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 6245 . 2  |-  ( R 
.\/  F )  =  ( R  .\/  (
( R  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) ) )
7269, 71syl6reqr 2462 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   joincjn 15789   meetcmee 15790   1.cp1 15884   Latclat 15891   OLcol 32173   Atomscatm 32262   HLchlt 32349   LHypclh 32982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-p1 15886  df-lat 15892  df-clat 15954  df-oposet 32175  df-ol 32177  df-oml 32178  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350  df-psubsp 32501  df-pmap 32502  df-padd 32794  df-lhyp 32986
This theorem is referenced by:  cdleme2  33227  cdleme3b  33228  cdleme3c  33229  cdleme5  33239  cdleme11  33269  cdleme12  33270  cdleme16c  33279  cdleme20g  33315  cdleme35a  33448  cdleme36a  33460
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