Metamath Proof Explorer

Theorem zrhpsgnevpm

Description: The sign of an even permutation embedded into a ring is the multiplicative neutral element of the ring. (Contributed by SO, 9-Jul-2018)

Ref Expression
Hypotheses zrhpsgnevpm.y 
|- Y = ( ZRHom ` R )
zrhpsgnevpm.s 
|- S = ( pmSgn ` N )
zrhpsgnevpm.o 
|- .1. = ( 1r ` R )
Assertion zrhpsgnevpm 
|- ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> ( ( Y o. S ) ` F ) = .1. )

Proof

Step Hyp Ref Expression
1 zrhpsgnevpm.y 
 |-  Y = ( ZRHom ` R )
2 zrhpsgnevpm.s 
 |-  S = ( pmSgn ` N )
3 zrhpsgnevpm.o 
 |-  .1. = ( 1r ` R )
4 eqid 
 |-  ( SymGrp ` N ) = ( SymGrp ` N )
5 eqid 
 |-  ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) = ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } )
6 4 2 5 psgnghm2 
 |-  ( N e. Fin -> S e. ( ( SymGrp ` N ) GrpHom ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) ) )
7 eqid 
 |-  ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
8 eqid 
 |-  ( Base ` ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) ) = ( Base ` ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) )
9 7 8 ghmf 
 |-  ( S e. ( ( SymGrp ` N ) GrpHom ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) ) -> S : ( Base ` ( SymGrp ` N ) ) --> ( Base ` ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) ) )
10 6 9 syl 
 |-  ( N e. Fin -> S : ( Base ` ( SymGrp ` N ) ) --> ( Base ` ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) ) )
11 10 3ad2ant2 
 |-  ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> S : ( Base ` ( SymGrp ` N ) ) --> ( Base ` ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) ) )
12 4 7 evpmss 
 |-  ( pmEven ` N ) C_ ( Base ` ( SymGrp ` N ) )
13 12 sseli 
 |-  ( F e. ( pmEven ` N ) -> F e. ( Base ` ( SymGrp ` N ) ) )
14 13 3ad2ant3 
 |-  ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> F e. ( Base ` ( SymGrp ` N ) ) )
15 fvco3 
 |-  ( ( S : ( Base ` ( SymGrp ` N ) ) --> ( Base ` ( ( mulGrp ` CCfld ) |`s { 1 , -u 1 } ) ) /\ F e. ( Base ` ( SymGrp ` N ) ) ) -> ( ( Y o. S ) ` F ) = ( Y ` ( S ` F ) ) )
16 11 14 15 syl2anc 
 |-  ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> ( ( Y o. S ) ` F ) = ( Y ` ( S ` F ) ) )
17 4 7 2 psgnevpm 
 |-  ( ( N e. Fin /\ F e. ( pmEven ` N ) ) -> ( S ` F ) = 1 )
18 17 3adant1 
 |-  ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> ( S ` F ) = 1 )
19 18 fveq2d 
 |-  ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> ( Y ` ( S ` F ) ) = ( Y ` 1 ) )
20 1 3 zrh1 
 |-  ( R e. Ring -> ( Y ` 1 ) = .1. )
21 20 3ad2ant1 
 |-  ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> ( Y ` 1 ) = .1. )
22 16 19 21 3eqtrd 
 |-  ( ( R e. Ring /\ N e. Fin /\ F e. ( pmEven ` N ) ) -> ( ( Y o. S ) ` F ) = .1. )