Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ltrncoval Structured version   Unicode version

Theorem ltrncoval 33162
Description: Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)
Hypotheses
Ref Expression
ltrnel.l  |-  .<_  =  ( le `  K )
ltrnel.a  |-  A  =  ( Atoms `  K )
ltrnel.h  |-  H  =  ( LHyp `  K
)
ltrnel.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncoval  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  (
( F  o.  G
) `  P )  =  ( F `  ( G `  P ) ) )

Proof of Theorem ltrncoval
StepHypRef Expression
1 simp1 997 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2r 1024 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  G  e.  T )
3 eqid 2402 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
4 ltrnel.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 ltrnel.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
63, 4, 5ltrn1o 33141 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
71, 2, 6syl2anc 659 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  G : ( Base `  K
)
-1-1-onto-> ( Base `  K )
)
8 f1of 5799 . . 3  |-  ( G : ( Base `  K
)
-1-1-onto-> ( Base `  K )  ->  G : ( Base `  K ) --> ( Base `  K ) )
97, 8syl 17 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  G : ( Base `  K
) --> ( Base `  K
) )
10 ltrnel.a . . . 4  |-  A  =  ( Atoms `  K )
113, 10atbase 32307 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
12113ad2ant3 1020 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  P  e.  ( Base `  K
) )
13 fvco3 5926 . 2  |-  ( ( G : ( Base `  K ) --> ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  (
( F  o.  G
) `  P )  =  ( F `  ( G `  P ) ) )
149, 12, 13syl2anc 659 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  (
( F  o.  G
) `  P )  =  ( F `  ( G `  P ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    o. ccom 4827   -->wf 5565   -1-1-onto->wf1o 5568   ` cfv 5569   Basecbs 14841   lecple 14916   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-ats 32285  df-laut 33006  df-ldil 33121  df-ltrn 33122
This theorem is referenced by:  cdlemg41  33737  trlcoabs  33740  trlcoabs2N  33741  trlcolem  33745  cdlemg44  33752  cdlemi2  33838  cdlemk2  33851  cdlemk4  33853  cdlemk8  33857  dia2dimlem4  34087  dihjatcclem3  34440
  Copyright terms: Public domain W3C validator