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Theorem cdlemg7fvN 35297
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 991 . . 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 1028 . . . 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 1017 . . . 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 34812 . . . 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 1223 . . 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 1018 . . . 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 35280 . . . 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 1223 . . 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 1027 . . 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 1029 . . . . . . . 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 35272 . . . . . . . 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 1232 . . . . . . 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 6292 . . . . . 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 1060 . . . . . . 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 34710 . . . . . . 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 1223 . . . . . 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 2503 . . . . 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 6293 . . . 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 2506 . . 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 35272 . . 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 1232 . 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 6293 . 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 2503 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   lecple 14553   joincjn 15422   meetcmee 15423   Atomscatm 33937   HLchlt 34024   LHypclh 34657   LTrncltrn 34774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-riotaBAD 33633
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-undef 6994  df-map 7414  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172  df-lvols 34173  df-lines 34174  df-psubsp 34176  df-pmap 34177  df-padd 34469  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832
This theorem is referenced by:  cdlemg7aN  35298
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