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Theorem cdlemg4g 36755
Description: TODO: FIX COMMENT (Contributed by NM, 25-Apr-2013.)
Hypotheses
Ref Expression
cdlemg4.l  |-  .<_  =  ( le `  K )
cdlemg4.a  |-  A  =  ( Atoms `  K )
cdlemg4.h  |-  H  =  ( LHyp `  K
)
cdlemg4.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg4.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg4.j  |-  .\/  =  ( join `  K )
cdlemg4b.v  |-  V  =  ( R `  G
)
cdlemg4.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
cdlemg4g  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  ( ( Q  .\/  V )  ./\  ( P  .\/  Q ) ) )

Proof of Theorem cdlemg4g
StepHypRef Expression
1 cdlemg4.l . . 3  |-  .<_  =  ( le `  K )
2 cdlemg4.a . . 3  |-  A  =  ( Atoms `  K )
3 cdlemg4.h . . 3  |-  H  =  ( LHyp `  K
)
4 cdlemg4.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
5 cdlemg4.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
6 cdlemg4.j . . 3  |-  .\/  =  ( join `  K )
7 cdlemg4b.v . . 3  |-  V  =  ( R `  G
)
8 cdlemg4.m . . 3  |-  ./\  =  ( meet `  K )
91, 2, 3, 4, 5, 6, 7, 8cdlemg4f 36754 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  ( ( Q  .\/  V )  ./\  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
) ) )
10 simp1l 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  K  e.  HL )
11 simp1r 1019 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  W  e.  H )
12 simp21 1027 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
13 simp22l 1113 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  Q  e.  A )
14 eqid 2382 . . . . 5  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
151, 6, 8, 2, 3, 14cdleme0cp 36352 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )
)  ->  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
)  =  ( P 
.\/  Q ) )
1610, 11, 12, 13, 15syl22anc 1227 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( P  .\/  Q ) )
1716oveq2d 6212 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  (
( Q  .\/  V
)  ./\  ( P  .\/  ( ( P  .\/  Q )  ./\  W )
) )  =  ( ( Q  .\/  V
)  ./\  ( P  .\/  Q ) ) )
189, 17eqtrd 2423 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( G  e.  T  /\  -.  Q  .<_  ( P  .\/  V
)  /\  ( F `  ( G `  P
) )  =  P ) )  ->  ( F `  ( G `  Q ) )  =  ( ( Q  .\/  V )  ./\  ( P  .\/  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   lecple 14709   joincjn 15690   meetcmee 15691   Atomscatm 35401   HLchlt 35488   LHypclh 36121   LTrncltrn 36238   trLctrl 36296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-riotaBAD 35097
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-undef 6920  df-map 7340  df-preset 15674  df-poset 15692  df-plt 15705  df-lub 15721  df-glb 15722  df-join 15723  df-meet 15724  df-p0 15786  df-p1 15787  df-lat 15793  df-clat 15855  df-oposet 35314  df-ol 35316  df-oml 35317  df-covers 35404  df-ats 35405  df-atl 35436  df-cvlat 35460  df-hlat 35489  df-llines 35635  df-lplanes 35636  df-lvols 35637  df-lines 35638  df-psubsp 35640  df-pmap 35641  df-padd 35933  df-lhyp 36125  df-laut 36126  df-ldil 36241  df-ltrn 36242  df-trl 36297
This theorem is referenced by:  cdlemg4  36756
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