This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdlemg47a

Description: TODO: fix comment. TODO: Use this above in place of ( FP ) = P antecedents? (Contributed by NM, 5-Jun-2013)

Ref Expression
Hypotheses cdlemg46.b 
|- B = ( Base ` K )
cdlemg46.h 
|- H = ( LHyp ` K )
cdlemg46.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemg47a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> ( F o. G ) = ( G o. F ) )

Proof

Step Hyp Ref Expression
1 cdlemg46.b 
 |-  B = ( Base ` K )
2 cdlemg46.h 
 |-  H = ( LHyp ` K )
3 cdlemg46.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
5 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> G e. T )
6 1 2 3 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> G : B -1-1-onto-> B )
7 4 5 6 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> G : B -1-1-onto-> B )
8 f1of 
 |-  ( G : B -1-1-onto-> B -> G : B --> B )
9 7 8 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> G : B --> B )
10 fcoi1 
 |-  ( G : B --> B -> ( G o. ( _I |` B ) ) = G )
11 9 10 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> ( G o. ( _I |` B ) ) = G )
12 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> F = ( _I |` B ) )
13 12 coeq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> ( G o. F ) = ( G o. ( _I |` B ) ) )
14 12 coeq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> ( F o. G ) = ( ( _I |` B ) o. G ) )
15 fcoi2 
 |-  ( G : B --> B -> ( ( _I |` B ) o. G ) = G )
16 9 15 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> ( ( _I |` B ) o. G ) = G )
17 14 16 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> ( F o. G ) = G )
18 11 13 17 3eqtr4rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ F = ( _I |` B ) ) -> ( F o. G ) = ( G o. F ) )