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Theorem cdlemg7fvN 31421
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 957 . . 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 994 . . . 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 983 . . . 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 30936 . . . 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 1184 . . 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 984 . . . 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 31404 . . . 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 1184 . . 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 993 . . 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 995 . . . . . . . 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 31396 . . . . . . . 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 1193 . . . . . . 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 6096 . . . . . 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 1026 . . . . . . 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 30834 . . . . . . 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 1184 . . . . . 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 2468 . . . . 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 6097 . . . 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 2471 . . 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 31396 . . 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 1193 . 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 6097 . 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 2468 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Atomscatm 30061   HLchlt 30148   LHypclh 30781   LTrncltrn 30898
This theorem is referenced by:  cdlemg7aN  31422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956
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