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Theorem cdlemg15 36568
Description: Eliminate the  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) condition from cdlemg13 36564. TODO: FIX COMMENT (Contributed by NM, 25-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg15  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )

Proof of Theorem cdlemg15
StepHypRef Expression
1 simpl11 1071 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simpl12 1072 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simpl13 1073 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simpl2l 1049 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q ) )  ->  F  e.  T )
5 simpl2r 1050 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q ) )  ->  G  e.  T )
6 simpr 461 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q ) )  ->  ( ( F `  ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =  ( P 
.\/  Q ) )
7 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
11 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
12 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
137, 8, 9, 10, 11, 12cdlemg8 36543 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
141, 2, 3, 4, 5, 6, 13syl132anc 1246 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =  ( P  .\/  Q ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
15 simpl1 999 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
16 simpl2 1000 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )  ->  ( F  e.  T  /\  G  e.  T ) )
17 simpl3 1001 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )  ->  ( R `  F )  =  ( R `  G ) )
18 simpr 461 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )  ->  ( ( F `
 ( G `  P ) )  .\/  ( F `  ( G `
 Q ) ) )  =/=  ( P 
.\/  Q ) )
19 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
207, 8, 9, 10, 11, 12, 19cdlemg15a 36567 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( ( R `  F )  =  ( R `  G )  /\  (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
2115, 16, 17, 18, 20syl112anc 1232 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
2214, 21pm2.61dane 2775 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T
)  /\  ( R `  F )  =  ( R `  G ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   lecple 14810   joincjn 15791   meetcmee 15792   Atomscatm 35174   HLchlt 35261   LHypclh 35894   LTrncltrn 36011   trLctrl 36069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34870
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15775  df-poset 15793  df-plt 15806  df-lub 15822  df-glb 15823  df-join 15824  df-meet 15825  df-p0 15887  df-p1 15888  df-lat 15894  df-clat 15956  df-oposet 35087  df-ol 35089  df-oml 35090  df-covers 35177  df-ats 35178  df-atl 35209  df-cvlat 35233  df-hlat 35262  df-llines 35408  df-lplanes 35409  df-lvols 35410  df-lines 35411  df-psubsp 35413  df-pmap 35414  df-padd 35706  df-lhyp 35898  df-laut 35899  df-ldil 36014  df-ltrn 36015  df-trl 36070
This theorem is referenced by:  cdlemg16  36569  cdlemg39  36628
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