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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 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐶 ∈ 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 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐶 ∈ 𝐵 ) → 𝑊 ∈ Ring )
7 simp1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐶 ∈ 𝐵 ) → 𝑊 ∈ LMod )
8 1 2 6 7 asclghm ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐶 ∈ 𝐵 ) → 𝐴 ∈ ( 𝑅 GrpHom 𝑊 ) )
9 simp3 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐶 ∈ 𝐵 ) → 𝐶 ∈ 𝐵 )
10 3 4 5 ghminv ⊢ ( ( 𝐴 ∈ ( 𝑅 GrpHom 𝑊 ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐴 ‘ ( 𝐼 ‘ 𝐶 ) ) = ( 𝐽 ‘ ( 𝐴 ‘ 𝐶 ) ) )
11 10 eqcomd ⊢ ( ( 𝐴 ∈ ( 𝑅 GrpHom 𝑊 ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐽 ‘ ( 𝐴 ‘ 𝐶 ) ) = ( 𝐴 ‘ ( 𝐼 ‘ 𝐶 ) ) )
12 8 9 11 syl2anc ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐶 ∈ 𝐵 ) → ( 𝐽 ‘ ( 𝐴 ‘ 𝐶 ) ) = ( 𝐴 ‘ ( 𝐼 ‘ 𝐶 ) ) )