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Theorem cdlemg6d 30786
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 957 . . . . . 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 990 . . . . . 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 993 . . . . . 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 30774 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( P  .\/  V )  =  ( P  .\/  ( G `  P )
) )
121, 2, 3, 11syl3anc 1184 . . . . 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 4158 . . . 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 286 . . 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 685 . 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 30785 . 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 227 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   lecple 13456   joincjn 14321   Atomscatm 29429   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   trLctrl 30323
This theorem is referenced by:  cdlemg6e  30787
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-map 6949  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664  df-lvols 29665  df-lines 29666  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324
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