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Theorem cdlemg7fvN 33607
Description: Value of a translation composition in terms of an associated atom. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg7fv.b  |-  B  =  ( Base `  K
)
cdlemg7fv.l  |-  .<_  =  ( le `  K )
cdlemg7fv.j  |-  .\/  =  ( join `  K )
cdlemg7fv.m  |-  ./\  =  ( meet `  K )
cdlemg7fv.a  |-  A  =  ( Atoms `  K )
cdlemg7fv.h  |-  H  =  ( LHyp `  K
)
cdlemg7fv.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemg7fvN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  ( G `  X ) )  =  ( ( F `  ( G `  P ) )  .\/  ( X 
./\  W ) ) )

Proof of Theorem cdlemg7fvN
StepHypRef Expression
1 simp1 995 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp32 1032 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  G  e.  T )
3 simp2l 1021 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 cdlemg7fv.l . . . . 5  |-  .<_  =  ( le `  K )
5 cdlemg7fv.a . . . . 5  |-  A  =  ( Atoms `  K )
6 cdlemg7fv.h . . . . 5  |-  H  =  ( LHyp `  K
)
7 cdlemg7fv.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
84, 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 ) )
91, 2, 3, 8syl3anc 1228 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( G `  P
)  e.  A  /\  -.  ( G `  P
)  .<_  W ) )
10 simp2r 1022 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
11 cdlemg7fv.b . . . . 5  |-  B  =  ( Base `  K
)
124, 5, 6, 7, 11cdlemg7fvbwN 33590 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  G  e.  T )  ->  (
( G `  X
)  e.  B  /\  -.  ( G `  X
)  .<_  W ) )
131, 10, 2, 12syl3anc 1228 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( G `  X
)  e.  B  /\  -.  ( G `  X
)  .<_  W ) )
14 simp31 1031 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  F  e.  T )
15 simp33 1033 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( P  .\/  ( X  ./\  W ) )  =  X )
16 cdlemg7fv.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
17 cdlemg7fv.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
186, 7, 4, 16, 5, 17, 11cdlemg2fv 33582 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  X )  =  ( ( G `
 P )  .\/  ( X  ./\  W ) ) )
191, 3, 10, 2, 15, 18syl122anc 1237 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  X )  =  ( ( G `
 P )  .\/  ( X  ./\  W ) ) )
2019oveq1d 6247 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( G `  X
)  ./\  W )  =  ( ( ( G `  P ) 
.\/  ( X  ./\  W ) )  ./\  W
) )
21 simp2rl 1064 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  X  e.  B )
2211, 4, 16, 17, 5, 6lhpelim 33018 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( G `
 P )  e.  A  /\  -.  ( G `  P )  .<_  W )  /\  X  e.  B )  ->  (
( ( G `  P )  .\/  ( X  ./\  W ) ) 
./\  W )  =  ( X  ./\  W
) )
231, 9, 21, 22syl3anc 1228 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( ( G `  P )  .\/  ( X  ./\  W ) ) 
./\  W )  =  ( X  ./\  W
) )
2420, 23eqtrd 2441 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( G `  X
)  ./\  W )  =  ( X  ./\  W ) )
2524oveq2d 6248 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( G `  P
)  .\/  ( ( G `  X )  ./\  W ) )  =  ( ( G `  P )  .\/  ( X  ./\  W ) ) )
2625, 19eqtr4d 2444 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( G `  P
)  .\/  ( ( G `  X )  ./\  W ) )  =  ( G `  X
) )
276, 7, 4, 16, 5, 17, 11cdlemg2fv 33582 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( G `  P )  e.  A  /\  -.  ( G `  P ) 
.<_  W )  /\  (
( G `  X
)  e.  B  /\  -.  ( G `  X
)  .<_  W ) )  /\  ( F  e.  T  /\  ( ( G `  P ) 
.\/  ( ( G `
 X )  ./\  W ) )  =  ( G `  X ) ) )  ->  ( F `  ( G `  X ) )  =  ( ( F `  ( G `  P ) )  .\/  ( ( G `  X ) 
./\  W ) ) )
281, 9, 13, 14, 26, 27syl122anc 1237 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  ( G `  X ) )  =  ( ( F `  ( G `  P ) )  .\/  ( ( G `  X ) 
./\  W ) ) )
2924oveq2d 6248 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  (
( F `  ( G `  P )
)  .\/  ( ( G `  X )  ./\  W ) )  =  ( ( F `  ( G `  P ) )  .\/  ( X 
./\  W ) ) )
3028, 29eqtrd 2441 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( P  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  ( G `  X ) )  =  ( ( F `  ( G `  P ) )  .\/  ( X 
./\  W ) ) )
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   Basecbs 14731   lecple 14806   joincjn 15787   meetcmee 15788   Atomscatm 32245   HLchlt 32332   LHypclh 32965   LTrncltrn 33082
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  ax-riotaBAD 31941
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  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-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  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-undef 6957  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-llines 32479  df-lplanes 32480  df-lvols 32481  df-lines 32482  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:  cdlemg7aN  33608
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