Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemc6 Structured version   Unicode version

Theorem cdlemc6 35661
Description: Lemma for cdlemc 35662. (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 1021 . . . . 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 1116 . . . . 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 1118 . . . . 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 34832 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
71, 2, 3, 6syl3anc 1229 . . . 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 6297 . . 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 34828 . . . . 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 2443 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1211, 5atbase 34754 . . . . 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 34754 . . . . 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 15592 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  ( Q  ./\  ( Q  .\/  P ) )  =  Q )
1810, 13, 15, 17syl3anc 1229 . . 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 2484 . 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 997 . . . . . 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 1031 . . . . . 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 1030 . . . . . 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 999 . . . . . 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 2443 . . . . . . 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 35635 . . . . . 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 1233 . . . . 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 6297 . . . 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 34826 . . . . . 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 34690 . . . . 5  |-  ( ( K  e.  OL  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  ( 0. `  K ) )  =  Q )
3533, 13, 34syl2anc 661 . . . 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 2484 . . 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 6296 . . . 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 34831 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
391, 2, 3, 38syl3anc 1229 . . . . . 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 1022 . . . . . . 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 35462 . . . . . . 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 15556 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  e.  ( Base `  K ) )
4410, 39, 42, 43syl3anc 1229 . . . . 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 15563 . . . . 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 1229 . . . 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 34839 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
481, 2, 3, 47syl3anc 1229 . . . . . 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 35325 . . . . . 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 1242 . . . . 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 2443 . . . . . . . 8  |-  ( 1.
`  K )  =  ( 1. `  K
)
5224, 4, 51, 5, 26lhpjat1 35484 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  ( 1.
`  K ) )
531, 40, 21, 52syl21anc 1228 . . . . . 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 6297 . . . . 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 34692 . . . . . 6  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
5633, 39, 55syl2anc 661 . . . . 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 2488 . . . 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 2488 . . 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 6299 . 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 35646 . 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 2495 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   lecple 14581   joincjn 15447   meetcmee 15448   0.cp0 15541   1.cp1 15542   Latclat 15549   OLcol 34639   Atomscatm 34728   HLchlt 34815   LHypclh 35448   LTrncltrn 35565   trLctrl 35623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-map 7424  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-psubsp 34967  df-pmap 34968  df-padd 35260  df-lhyp 35452  df-laut 35453  df-ldil 35568  df-ltrn 35569  df-trl 35624
This theorem is referenced by:  cdlemc  35662
  Copyright terms: Public domain W3C validator