Metamath Proof Explorer


Theorem assaascl1

Description: The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019)

Ref Expression
Hypotheses assaascl0.a 𝐴 = ( algSc ‘ 𝑊 )
assaascl0.f 𝐹 = ( Scalar ‘ 𝑊 )
assaascl0.w ( 𝜑𝑊 ∈ AssAlg )
Assertion assaascl1 ( 𝜑 → ( 𝐴 ‘ ( 1r𝐹 ) ) = ( 1r𝑊 ) )

Proof

Step Hyp Ref Expression
1 assaascl0.a 𝐴 = ( algSc ‘ 𝑊 )
2 assaascl0.f 𝐹 = ( Scalar ‘ 𝑊 )
3 assaascl0.w ( 𝜑𝑊 ∈ AssAlg )
4 assalmod ( 𝑊 ∈ AssAlg → 𝑊 ∈ LMod )
5 3 4 syl ( 𝜑𝑊 ∈ LMod )
6 assaring ( 𝑊 ∈ AssAlg → 𝑊 ∈ Ring )
7 3 6 syl ( 𝜑𝑊 ∈ Ring )
8 1 2 5 7 ascl1 ( 𝜑 → ( 𝐴 ‘ ( 1r𝐹 ) ) = ( 1r𝑊 ) )