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Theorem ltrncoval 35431
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 1061 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( K e. HL /\ W e. H ) )
2 simp2r 1088 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> G e. T )
3 eqid 2622 . . . . 5 |- ( Base ` K ) = ( Base ` K )
4 ltrnel.h . . . . 5 |- H = ( LHyp ` K )
5 ltrnel.t . . . . 5 |- T = ( ( LTrn ` K ) ` W )
63, 4, 5ltrn1o 35410 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> G : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
71, 2, 6syl2anc 693 . . 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 6137 . . 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 34576 . . 3 |- ( P e. A -> P e. ( Base ` K ) )
12113ad2ant3 1084 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> P e. ( Base ` K ) )
13 fvco3 6275 . 2 |- ( ( G : ( Base ` K ) --> ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( ( F o. G ) ` P ) = ( F ` ( G ` P ) ) )
149, 12, 13syl2anc 693 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Basecbs 15857   lecple 15948   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ats 34554  df-laut 35275  df-ldil 35390  df-ltrn 35391
This theorem is referenced by:  cdlemg41  36006  trlcoabs  36009  trlcoabs2N  36010  trlcolem  36014  cdlemg44  36021  cdlemi2  36107  cdlemk2  36120  cdlemk4  36122  cdlemk8  36126  dia2dimlem4  36356  dihjatcclem3  36709
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