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Metamath Proof Explorer

Theorem ltrncoval

Description: Two ways to express value of translation composition. (Contributed by NM, 31-May-2013)

Ref Expression
Hypotheses ltrnel.l 
|- .<_ = ( le ` K )
ltrnel.a 
|- A = ( Atoms ` K )
ltrnel.h 
|- H = ( LHyp ` K )
ltrnel.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrncoval 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( ( F o. G ) ` P ) = ( F ` ( G ` P ) ) )

Proof

Step Hyp Ref Expression
1 ltrnel.l 
 |-  .<_ = ( le ` K )
2 ltrnel.a 
 |-  A = ( Atoms ` K )
3 ltrnel.h 
 |-  H = ( LHyp ` K )
4 ltrnel.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( K e. HL /\ W e. H ) )
6 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> G e. T )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 3 4 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> G : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
9 5 6 8 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> G : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
10 f1of 
 |-  ( G : ( Base ` K ) -1-1-onto-> ( Base ` K ) -> G : ( Base ` K ) --> ( Base ` K ) )
11 9 10 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> G : ( Base ` K ) --> ( Base ` K ) )
12 7 2 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
13 12 3ad2ant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> P e. ( Base ` K ) )
14 fvco3 
 |-  ( ( G : ( Base ` K ) --> ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( ( F o. G ) ` P ) = ( F ` ( G ` P ) ) )
15 11 13 14 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( ( F o. G ) ` P ) = ( F ` ( G ` P ) ) )