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Theorem ltrncoval 34816
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 991 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2r 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  G  e.  T )
3 eqid 2460 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
4 ltrnel.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 ltrnel.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
63, 4, 5ltrn1o 34795 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  G :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
71, 2, 6syl2anc 661 . . 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 5807 . . 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 33961 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
12113ad2ant3 1014 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  e.  A )  ->  P  e.  ( Base `  K
) )
13 fvco3 5935 . 2  |-  ( ( G : ( Base `  K ) --> ( Base `  K )  /\  P  e.  ( Base `  K
) )  ->  (
( F  o.  G
) `  P )  =  ( F `  ( G `  P ) ) )
149, 12, 13syl2anc 661 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762    o. ccom 4996   -->wf 5575   -1-1-onto->wf1o 5578   ` cfv 5579   Basecbs 14479   lecple 14551   Atomscatm 33935   HLchlt 34022   LHypclh 34655   LTrncltrn 34772
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 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-ats 33939  df-laut 34660  df-ldil 34775  df-ltrn 34776
This theorem is referenced by:  cdlemg41  35389  trlcoabs  35392  trlcoabs2N  35393  trlcolem  35397  cdlemg44  35404  cdlemi2  35490  cdlemk2  35503  cdlemk4  35505  cdlemk8  35509  dia2dimlem4  35739  dihjatcclem3  36092
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