Metamath Proof Explorer


Theorem asclinvg

Description: The group inverse (negation) of a lifted scalar is the lifted negation of the scalar. (Contributed by AV, 2-Sep-2019)

Ref Expression
Hypotheses asclinvg.a
|- A = ( algSc ` W )
asclinvg.r
|- R = ( Scalar ` W )
asclinvg.k
|- B = ( Base ` R )
asclinvg.i
|- I = ( invg ` R )
asclinvg.j
|- J = ( invg ` W )
Assertion asclinvg
|- ( ( W e. LMod /\ W e. Ring /\ C e. B ) -> ( J ` ( A ` C ) ) = ( A ` ( I ` C ) ) )

Proof

Step Hyp Ref Expression
1 asclinvg.a
 |-  A = ( algSc ` W )
2 asclinvg.r
 |-  R = ( Scalar ` W )
3 asclinvg.k
 |-  B = ( Base ` R )
4 asclinvg.i
 |-  I = ( invg ` R )
5 asclinvg.j
 |-  J = ( invg ` W )
6 simp2
 |-  ( ( W e. LMod /\ W e. Ring /\ C e. B ) -> W e. Ring )
7 simp1
 |-  ( ( W e. LMod /\ W e. Ring /\ C e. B ) -> W e. LMod )
8 1 2 6 7 asclghm
 |-  ( ( W e. LMod /\ W e. Ring /\ C e. B ) -> A e. ( R GrpHom W ) )
9 simp3
 |-  ( ( W e. LMod /\ W e. Ring /\ C e. B ) -> C e. B )
10 3 4 5 ghminv
 |-  ( ( A e. ( R GrpHom W ) /\ C e. B ) -> ( A ` ( I ` C ) ) = ( J ` ( A ` C ) ) )
11 10 eqcomd
 |-  ( ( A e. ( R GrpHom W ) /\ C e. B ) -> ( J ` ( A ` C ) ) = ( A ` ( I ` C ) ) )
12 8 9 11 syl2anc
 |-  ( ( W e. LMod /\ W e. Ring /\ C e. B ) -> ( J ` ( A ` C ) ) = ( A ` ( I ` C ) ) )