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Theorem ltrncoval 31116
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 958 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2r 985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  G  e.  T )
3 eqid 2443 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
4 ltrnel.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 ltrnel.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
63, 4, 5ltrn1o 31095 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
71, 2, 6syl2anc 644 . . 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 5709 . . 3  |-  ( G : ( Base `  K
)
-1-1-onto-> ( Base `  K )  ->  G : ( Base `  K ) --> ( Base `  K ) )
97, 8syl 16 . 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 30261 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
12113ad2ant3 981 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  P  e.  ( Base `  K
) )
13 fvco3 5836 . 2  |-  ( ( G : ( Base `  K ) --> ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  (
( F  o.  G
) `  P )  =  ( F `  ( G `  P ) ) )
149, 12, 13syl2anc 644 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728    o. ccom 4917   -->wf 5485   -1-1-onto->wf1o 5488   ` cfv 5489   Basecbs 13507   lecple 13574   Atomscatm 30235   HLchlt 30322   LHypclh 30955   LTrncltrn 31072
This theorem is referenced by:  cdlemg41  31689  trlcoabs  31692  trlcoabs2N  31693  trlcolem  31697  cdlemg44  31704  cdlemi2  31790  cdlemk2  31803  cdlemk4  31805  cdlemk8  31809  dia2dimlem4  32039  dihjatcclem3  32392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-map 7056  df-ats 30239  df-laut 30960  df-ldil 31075  df-ltrn 31076
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