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Theorem cdlemg6d 31432
Description: TODO: FIX COMMENT (Contributed by NM, 27-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
)
Assertion
Ref Expression
cdlemg6d  |-  ( ( ( 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 ) )  ->  ( (
( r  e.  A  /\  -.  r  .<_  W )  /\  -.  r  .<_  ( P  .\/  ( G `
 P ) ) )  ->  ( F `  ( G `  Q
) )  =  Q ) )
Distinct variable groups:    A, r    F, r    G, r    H, r    .\/ , r    K, r    .<_ , r    P, r    Q, r    T, r    V, r    W, r
Allowed substitution hint:    R( r)

Proof of Theorem cdlemg6d
StepHypRef Expression
1 simp1 955 . . . . . 6  |-  ( ( ( 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  /\  W  e.  H ) )
2 simp21 988 . . . . . 6  |-  ( ( ( 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 ) )
3 simp31 991 . . . . . 6  |-  ( ( ( 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 ) )  ->  G  e.  T )
4 cdlemg4.l . . . . . . 7  |-  .<_  =  ( le `  K )
5 cdlemg4.a . . . . . . 7  |-  A  =  ( Atoms `  K )
6 cdlemg4.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
7 cdlemg4.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemg4.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
9 cdlemg4.j . . . . . . 7  |-  .\/  =  ( join `  K )
10 cdlemg4b.v . . . . . . 7  |-  V  =  ( R `  G
)
114, 5, 6, 7, 8, 9, 10cdlemg4b1 31420 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( P  .\/  V )  =  ( P  .\/  ( G `  P )
) )
121, 2, 3, 11syl3anc 1182 . . . . 5  |-  ( ( ( 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  .\/  V )  =  ( P  .\/  ( G `
 P ) ) )
1312breq2d 4051 . . . 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 ) )  ->  ( r  .<_  ( P  .\/  V
)  <->  r  .<_  ( P 
.\/  ( G `  P ) ) ) )
1413notbid 285 . . 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 ) )  ->  ( -.  r  .<_  ( P  .\/  V )  <->  -.  r  .<_  ( P  .\/  ( G `
 P ) ) ) )
1514anbi2d 684 . 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 ) )  ->  ( (
( r  e.  A  /\  -.  r  .<_  W )  /\  -.  r  .<_  ( P  .\/  V ) )  <->  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  -.  r  .<_  ( P 
.\/  ( G `  P ) ) ) ) )
164, 5, 6, 7, 8, 9, 10cdlemg6c 31431 . 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 ) )  ->  ( (
( r  e.  A  /\  -.  r  .<_  W )  /\  -.  r  .<_  ( P  .\/  V ) )  ->  ( F `  ( G `  Q
) )  =  Q ) )
1715, 16sylbird 226 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 ) )  ->  ( (
( r  e.  A  /\  -.  r  .<_  W )  /\  -.  r  .<_  ( P  .\/  ( G `
 P ) ) )  ->  ( F `  ( G `  Q
) )  =  Q ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
This theorem is referenced by:  cdlemg6e  31433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
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