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Theorem cdlemd6 33847
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)
Hypotheses
Ref Expression
cdlemd4.l  |-  .<_  =  ( le `  K )
cdlemd4.j  |-  .\/  =  ( join `  K )
cdlemd4.a  |-  A  =  ( Atoms `  K )
cdlemd4.h  |-  H  =  ( LHyp `  K
)
cdlemd4.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemd6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( F `  Q
)  =  ( G `
 Q ) )

Proof of Theorem cdlemd6
StepHypRef Expression
1 simp3 990 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( F `  P
)  =  ( G `
 P ) )
21oveq2d 6107 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( P  .\/  ( F `  P )
)  =  ( P 
.\/  ( G `  P ) ) )
32oveq1d 6106 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  =  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W ) )
4 simp1l 1012 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
5 simp1rl 1053 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  ->  F  e.  T )
6 simp21 1021 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
7 cdlemd4.l . . . . . . 7  |-  .<_  =  ( le `  K )
8 cdlemd4.j . . . . . . 7  |-  .\/  =  ( join `  K )
9 eqid 2443 . . . . . . 7  |-  ( meet `  K )  =  (
meet `  K )
10 cdlemd4.a . . . . . . 7  |-  A  =  ( Atoms `  K )
11 cdlemd4.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
12 cdlemd4.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
13 eqid 2443 . . . . . . 7  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
147, 8, 9, 10, 11, 12, 13trlval2 33807 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( trL `  K
) `  W ) `  F )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) W ) )
154, 5, 6, 14syl3anc 1218 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( ( ( trL `  K ) `  W
) `  F )  =  ( ( P 
.\/  ( F `  P ) ) (
meet `  K ) W ) )
16 simp1rr 1054 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  ->  G  e.  T )
177, 8, 9, 10, 11, 12, 13trlval2 33807 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( trL `  K
) `  W ) `  G )  =  ( ( P  .\/  ( G `  P )
) ( meet `  K
) W ) )
184, 16, 6, 17syl3anc 1218 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( ( ( trL `  K ) `  W
) `  G )  =  ( ( P 
.\/  ( G `  P ) ) (
meet `  K ) W ) )
193, 15, 183eqtr4d 2485 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( ( ( trL `  K ) `  W
) `  F )  =  ( ( ( trL `  K ) `
 W ) `  G ) )
2019oveq2d 6107 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( Q  .\/  (
( ( trL `  K
) `  W ) `  F ) )  =  ( Q  .\/  (
( ( trL `  K
) `  W ) `  G ) ) )
211oveq1d 6106 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( ( F `  P )  .\/  (
( P  .\/  Q
) ( meet `  K
) W ) )  =  ( ( G `
 P )  .\/  ( ( P  .\/  Q ) ( meet `  K
) W ) ) )
2220, 21oveq12d 6109 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( ( Q  .\/  ( ( ( trL `  K ) `  W
) `  F )
) ( meet `  K
) ( ( F `
 P )  .\/  ( ( P  .\/  Q ) ( meet `  K
) W ) ) )  =  ( ( Q  .\/  ( ( ( trL `  K
) `  W ) `  G ) ) (
meet `  K )
( ( G `  P )  .\/  (
( P  .\/  Q
) ( meet `  K
) W ) ) ) )
23 simp22 1022 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
24 simp23 1023 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  ->  -.  Q  .<_  ( P 
.\/  ( F `  P ) ) )
257, 8, 9, 10, 11, 12, 13cdlemc 33841 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  -.  Q  .<_  ( P  .\/  ( F `  P )
) )  ->  ( F `  Q )  =  ( ( Q 
.\/  ( ( ( trL `  K ) `
 W ) `  F ) ) (
meet `  K )
( ( F `  P )  .\/  (
( P  .\/  Q
) ( meet `  K
) W ) ) ) )
264, 5, 6, 23, 24, 25syl131anc 1231 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( F `  Q
)  =  ( ( Q  .\/  ( ( ( trL `  K
) `  W ) `  F ) ) (
meet `  K )
( ( F `  P )  .\/  (
( P  .\/  Q
) ( meet `  K
) W ) ) ) )
27 oveq2 6099 . . . . . . 7  |-  ( ( F `  P )  =  ( G `  P )  ->  ( P  .\/  ( F `  P ) )  =  ( P  .\/  ( G `  P )
) )
2827breq2d 4304 . . . . . 6  |-  ( ( F `  P )  =  ( G `  P )  ->  ( Q  .<_  ( P  .\/  ( F `  P ) )  <->  Q  .<_  ( P 
.\/  ( G `  P ) ) ) )
2928notbid 294 . . . . 5  |-  ( ( F `  P )  =  ( G `  P )  ->  ( -.  Q  .<_  ( P 
.\/  ( F `  P ) )  <->  -.  Q  .<_  ( P  .\/  ( G `  P )
) ) )
3029biimpd 207 . . . 4  |-  ( ( F `  P )  =  ( G `  P )  ->  ( -.  Q  .<_  ( P 
.\/  ( F `  P ) )  ->  -.  Q  .<_  ( P 
.\/  ( G `  P ) ) ) )
311, 24, 30sylc 60 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  ->  -.  Q  .<_  ( P 
.\/  ( G `  P ) ) )
327, 8, 9, 10, 11, 12, 13cdlemc 33841 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  -.  Q  .<_  ( P  .\/  ( G `  P )
) )  ->  ( G `  Q )  =  ( ( Q 
.\/  ( ( ( trL `  K ) `
 W ) `  G ) ) (
meet `  K )
( ( G `  P )  .\/  (
( P  .\/  Q
) ( meet `  K
) W ) ) ) )
334, 16, 6, 23, 31, 32syl131anc 1231 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( G `  Q
)  =  ( ( Q  .\/  ( ( ( trL `  K
) `  W ) `  G ) ) (
meet `  K )
( ( G `  P )  .\/  (
( P  .\/  Q
) ( meet `  K
) W ) ) ) )
3422, 26, 333eqtr4d 2485 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  -.  Q  .<_  ( P  .\/  ( F `  P ) ) )  /\  ( F `  P )  =  ( G `  P ) )  -> 
( F `  Q
)  =  ( G `
 Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   lecple 14245   joincjn 15114   meetcmee 15115   Atomscatm 32908   HLchlt 32995   LHypclh 33628   LTrncltrn 33745   trLctrl 33802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-map 7216  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803
This theorem is referenced by:  cdlemd7  33848
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