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Theorem cdlemg4b2 33593
Description: TODO: FIX COMMENT (Contributed by NM, 24-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
cdlemg4b2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( P  .\/  ( G `  P ) ) )

Proof of Theorem cdlemg4b2
StepHypRef Expression
1 cdlemg4b.v . . . 4  |-  V  =  ( R `  G
)
2 cdlemg4.l . . . . . 6  |-  .<_  =  ( le `  K )
3 cdlemg4.j . . . . . 6  |-  .\/  =  ( join `  K )
4 eqid 2400 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
5 cdlemg4.a . . . . . 6  |-  A  =  ( Atoms `  K )
6 cdlemg4.h . . . . . 6  |-  H  =  ( LHyp `  K
)
7 cdlemg4.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemg4.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
92, 3, 4, 5, 6, 7, 8trlval2 33145 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
) ( meet `  K
) W ) )
1093com23 1201 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( R `  G )  =  ( ( P 
.\/  ( G `  P ) ) (
meet `  K ) W ) )
111, 10syl5eq 2453 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  V  =  ( ( P 
.\/  ( G `  P ) ) (
meet `  K ) W ) )
1211oveq2d 6248 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( ( G `
 P )  .\/  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W ) ) )
13 simp1 995 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  ( K  e.  HL  /\  W  e.  H ) )
14 simp2l 1021 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  P  e.  A )
152, 5, 6, 7ltrnel 33120 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  e.  A  /\  -.  ( G `  P )  .<_  W ) )
16153com23 1201 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  e.  A  /\  -.  ( G `  P
)  .<_  W ) )
17 eqid 2400 . . . 4  |-  ( ( P  .\/  ( G `
 P ) ) ( meet `  K
) W )  =  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W )
182, 3, 4, 5, 6, 17cdleme0cq 33197 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( ( G `  P )  e.  A  /\  -.  ( G `  P ) 
.<_  W ) ) )  ->  ( ( G `
 P )  .\/  ( ( P  .\/  ( G `  P ) ) ( meet `  K
) W ) )  =  ( P  .\/  ( G `  P ) ) )
1913, 14, 16, 18syl12anc 1226 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  ( ( P  .\/  ( G `  P ) ) (
meet `  K ) W ) )  =  ( P  .\/  ( G `  P )
) )
2012, 19eqtrd 2441 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  G  e.  T )  ->  (
( G `  P
)  .\/  V )  =  ( P  .\/  ( G `  P ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   lecple 14806   joincjn 15787   meetcmee 15788   Atomscatm 32245   HLchlt 32332   LHypclh 32965   LTrncltrn 33082   trLctrl 33140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-map 7377  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-psubsp 32484  df-pmap 32485  df-padd 32777  df-lhyp 32969  df-laut 32970  df-ldil 33085  df-ltrn 33086  df-trl 33141
This theorem is referenced by:  cdlemg4b12  33594  cdlemg4c  33595
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