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Theorem cdlemd6 30685
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 959 . . . . . . 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 6056 . . . . . 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 6055 . . . . 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 981 . . . . . 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 1022 . . . . . 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 990 . . . . . 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 2404 . . . . . . 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 2404 . . . . . . 7  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
147, 8, 9, 10, 11, 12, 13trlval2 30645 . . . . . 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 1184 . . . . 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 1023 . . . . . 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 30645 . . . . . 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 1184 . . . . 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 2446 . . . 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 6056 . . 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 6055 . . 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 6058 . 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 991 . . 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 992 . . 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 30679 . . 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 1197 . 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 6048 . . . . . . 7  |-  ( ( F `  P )  =  ( G `  P )  ->  ( P  .\/  ( F `  P ) )  =  ( P  .\/  ( G `  P )
) )
2827breq2d 4184 . . . . . 6  |-  ( ( F `  P )  =  ( G `  P )  ->  ( Q  .<_  ( P  .\/  ( F `  P ) )  <->  Q  .<_  ( P 
.\/  ( G `  P ) ) ) )
2928notbid 286 . . . . 5  |-  ( ( F `  P )  =  ( G `  P )  ->  ( -.  Q  .<_  ( P 
.\/  ( F `  P ) )  <->  -.  Q  .<_  ( P  .\/  ( G `  P )
) ) )
3029biimpd 199 . . . 4  |-  ( ( F `  P )  =  ( G `  P )  ->  ( -.  Q  .<_  ( P 
.\/  ( F `  P ) )  ->  -.  Q  .<_  ( P 
.\/  ( G `  P ) ) ) )
311, 24, 30sylc 58 . . 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 30679 . . 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 1197 . 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 2446 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  cdlemd7  30686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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