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Theorem cdlemc6 30920
Description: Lemma for cdlemc 30921. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
cdlemc3.l  |-  .<_  =  ( le `  K )
cdlemc3.j  |-  .\/  =  ( join `  K )
cdlemc3.m  |-  ./\  =  ( meet `  K )
cdlemc3.a  |-  A  =  ( Atoms `  K )
cdlemc3.h  |-  H  =  ( LHyp `  K
)
cdlemc3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemc3.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemc6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  Q )  =  ( ( Q 
.\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) ) )

Proof of Theorem cdlemc6
StepHypRef Expression
1 simp1l 981 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  HL )
2 simp22l 1076 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  P  e.  A )
3 simp23l 1078 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  Q  e.  A )
4 cdlemc3.j . . . . . 6  |-  .\/  =  ( join `  K )
5 cdlemc3.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5hlatjcom 30092 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
71, 2, 3, 6syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
87oveq2d 6089 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( Q  ./\  ( P  .\/  Q ) )  =  ( Q  ./\  ( Q  .\/  P ) ) )
9 hllat 30088 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
101, 9syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  Lat )
11 eqid 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1211, 5atbase 30014 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
133, 12syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  Q  e.  ( Base `  K
) )
1411, 5atbase 30014 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
152, 14syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  P  e.  ( Base `  K
) )
16 cdlemc3.m . . . . 5  |-  ./\  =  ( meet `  K )
1711, 4, 16latabs2 14509 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  ( Q  ./\  ( Q  .\/  P ) )  =  Q )
1810, 13, 15, 17syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( Q  ./\  ( Q  .\/  P ) )  =  Q )
198, 18eqtrd 2467 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( Q  ./\  ( P  .\/  Q ) )  =  Q )
20 simp1 957 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
21 simp22 991 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
22 simp21 990 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  F  e.  T )
23 simp3 959 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  P )  =  P )
24 cdlemc3.l . . . . . . 7  |-  .<_  =  ( le `  K )
25 eqid 2435 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
26 cdlemc3.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
27 cdlemc3.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
28 cdlemc3.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
2924, 25, 5, 26, 27, 28trl0 30894 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( 0. `  K ) )
3020, 21, 22, 23, 29syl112anc 1188 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( R `  F )  =  ( 0. `  K ) )
3130oveq2d 6089 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( Q  .\/  ( R `  F ) )  =  ( Q  .\/  ( 0. `  K ) ) )
32 hlol 30086 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
331, 32syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  OL )
3411, 4, 25olj01 29950 . . . . 5  |-  ( ( K  e.  OL  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  ( 0. `  K ) )  =  Q )
3533, 13, 34syl2anc 643 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( Q  .\/  ( 0. `  K ) )  =  Q )
3631, 35eqtrd 2467 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( Q  .\/  ( R `  F ) )  =  Q )
3723oveq1d 6088 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  (
( F `  P
)  .\/  ( ( P  .\/  Q )  ./\  W ) )  =  ( P  .\/  ( ( P  .\/  Q ) 
./\  W ) ) )
3811, 4, 5hlatjcl 30091 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
391, 2, 3, 38syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
40 simp1r 982 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  W  e.  H )
4111, 26lhpbase 30722 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
4240, 41syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  W  e.  ( Base `  K
) )
4311, 16latmcl 14472 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
4410, 39, 42, 43syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  (
( P  .\/  Q
)  ./\  W )  e.  ( Base `  K
) )
4511, 4latjcom 14480 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  ./\  W )  e.  ( Base `  K
) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( ( P  .\/  Q )  ./\  W )  .\/  P ) )
4610, 15, 44, 45syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( ( P  .\/  Q )  ./\  W )  .\/  P ) )
4724, 4, 5hlatlej1 30099 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
481, 2, 3, 47syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  P  .<_  ( P  .\/  Q
) )
4911, 24, 4, 16, 5atmod2i1 30585 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  Q
) )  ->  (
( ( P  .\/  Q )  ./\  W )  .\/  P )  =  ( ( P  .\/  Q
)  ./\  ( W  .\/  P ) ) )
501, 2, 39, 42, 48, 49syl131anc 1197 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  (
( ( P  .\/  Q )  ./\  W )  .\/  P )  =  ( ( P  .\/  Q
)  ./\  ( W  .\/  P ) ) )
51 eqid 2435 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
5224, 4, 51, 5, 26lhpjat1 30744 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  ( 1.
`  K ) )
531, 40, 21, 52syl21anc 1183 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( W  .\/  P )  =  ( 1. `  K
) )
5453oveq2d 6089 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  (
( P  .\/  Q
)  ./\  ( W  .\/  P ) )  =  ( ( P  .\/  Q )  ./\  ( 1. `  K ) ) )
5511, 16, 51olm11 29952 . . . . . 6  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
5633, 39, 55syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
5750, 54, 563eqtrd 2471 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  (
( ( P  .\/  Q )  ./\  W )  .\/  P )  =  ( P  .\/  Q ) )
5837, 46, 573eqtrd 2471 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  (
( F `  P
)  .\/  ( ( P  .\/  Q )  ./\  W ) )  =  ( P  .\/  Q ) )
5936, 58oveq12d 6091 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  (
( Q  .\/  ( R `  F )
)  ./\  ( ( F `  P )  .\/  ( ( P  .\/  Q )  ./\  W )
) )  =  ( Q  ./\  ( P  .\/  Q ) ) )
6024, 5, 26, 27ltrnateq 30905 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  Q )  =  Q )
6119, 59, 603eqtr4rd 2478 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  Q )  =  ( ( Q 
.\/  ( R `  F ) )  ./\  ( ( F `  P )  .\/  (
( P  .\/  Q
)  ./\  W )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   0.cp0 14458   1.cp1 14459   Latclat 14466   OLcol 29899   Atomscatm 29988   HLchlt 30075   LHypclh 30708   LTrncltrn 30825   trLctrl 30882
This theorem is referenced by:  cdlemc  30921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712  df-laut 30713  df-ldil 30828  df-ltrn 30829  df-trl 30883
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