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Theorem cdlemc6 31091
Description: Lemma for cdlemc 31092. (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 982 . . . . 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 1077 . . . . 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 1079 . . . . 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 30263 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
71, 2, 3, 6syl3anc 1185 . . . 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 6126 . . 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 30259 . . . . 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 2442 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1211, 5atbase 30185 . . . . 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 30185 . . . . 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 14548 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  ( Q  ./\  ( Q  .\/  P ) )  =  Q )
1810, 13, 15, 17syl3anc 1185 . . 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 2474 . 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 958 . . . . . 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 992 . . . . . 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 991 . . . . . 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 960 . . . . . 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 2442 . . . . . . 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 31065 . . . . . 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 1189 . . . . 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 6126 . . . 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 30257 . . . . . 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 30121 . . . . 5  |-  ( ( K  e.  OL  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  ( 0. `  K ) )  =  Q )
3533, 13, 34syl2anc 644 . . . 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 2474 . . 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 6125 . . . 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 30262 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
391, 2, 3, 38syl3anc 1185 . . . . . 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 983 . . . . . . 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 30893 . . . . . . 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 14511 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
4410, 39, 42, 43syl3anc 1185 . . . . 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 14519 . . . . 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 1185 . . . 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 30270 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
481, 2, 3, 47syl3anc 1185 . . . . . 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 30756 . . . . . 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 1198 . . . . 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 2442 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
5224, 4, 51, 5, 26lhpjat1 30915 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  ( 1.
`  K ) )
531, 40, 21, 52syl21anc 1184 . . . . . 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 6126 . . . . 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 30123 . . . . . 6  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
5633, 39, 55syl2anc 644 . . . . 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 2478 . . . 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 2478 . . 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 6128 . 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 31076 . 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 2485 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   Basecbs 13500   lecple 13567   joincjn 14432   meetcmee 14433   0.cp0 14497   1.cp1 14498   Latclat 14505   OLcol 30070   Atomscatm 30159   HLchlt 30246   LHypclh 30879   LTrncltrn 30996   trLctrl 31053
This theorem is referenced by:  cdlemc  31092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-map 7049  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-psubsp 30398  df-pmap 30399  df-padd 30691  df-lhyp 30883  df-laut 30884  df-ldil 30999  df-ltrn 31000  df-trl 31054
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