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Metamath Proof Explorer

Theorem ltrncoidN

Description: Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analogue of vector subtraction. (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses ltrn1o.b 
|- B = ( Base ` K )
ltrn1o.h 
|- H = ( LHyp ` K )
ltrn1o.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrncoidN 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( F o. `' G ) = ( _I |` B ) <-> F = G ) )

Proof

Step Hyp Ref Expression
1 ltrn1o.b 
 |-  B = ( Base ` K )
2 ltrn1o.h 
 |-  H = ( LHyp ` K )
3 ltrn1o.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( K e. HL /\ W e. H ) )
5 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _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 o. `' G ) = ( _I |` B ) ) -> G : B -1-1-onto-> B )
8 f1ococnv1 
 |-  ( G : B -1-1-onto-> B -> ( `' G o. G ) = ( _I |` B ) )
9 7 8 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( `' G o. G ) = ( _I |` B ) )
10 9 coeq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( F o. ( `' G o. G ) ) = ( F o. ( _I |` B ) ) )
11 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F e. T )
12 1 2 3 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : B -1-1-onto-> B )
13 4 11 12 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F : B -1-1-onto-> B )
14 f1of 
 |-  ( F : B -1-1-onto-> B -> F : B --> B )
15 fcoi1 
 |-  ( F : B --> B -> ( F o. ( _I |` B ) ) = F )
16 13 14 15 3syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( F o. ( _I |` B ) ) = F )
17 10 16 eqtr2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F = ( F o. ( `' G o. G ) ) )
18 coass 
 |-  ( ( F o. `' G ) o. G ) = ( F o. ( `' G o. G ) )
19 17 18 eqtr4di 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F = ( ( F o. `' G ) o. G ) )
20 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( F o. `' G ) = ( _I |` B ) )
21 20 coeq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( ( F o. `' G ) o. G ) = ( ( _I |` B ) o. G ) )
22 f1of 
 |-  ( G : B -1-1-onto-> B -> G : B --> B )
23 fcoi2 
 |-  ( G : B --> B -> ( ( _I |` B ) o. G ) = G )
24 7 22 23 3syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( ( _I |` B ) o. G ) = G )
25 21 24 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> ( ( F o. `' G ) o. G ) = G )
26 19 25 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( F o. `' G ) = ( _I |` B ) ) -> F = G )
27 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> F = G )
28 27 coeq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> ( F o. `' G ) = ( G o. `' G ) )
29 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> ( K e. HL /\ W e. H ) )
30 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> G e. T )
31 29 30 6 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> G : B -1-1-onto-> B )
32 f1ococnv2 
 |-  ( G : B -1-1-onto-> B -> ( G o. `' G ) = ( _I |` B ) )
33 31 32 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> ( G o. `' G ) = ( _I |` B ) )
34 28 33 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ F = G ) -> ( F o. `' G ) = ( _I |` B ) )
35 26 34 impbida 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( ( F o. `' G ) = ( _I |` B ) <-> F = G ) )