Metamath Proof Explorer


Theorem psr1vsca

Description: Value of scalar multiplication in a univariate power series ring. (Contributed by Stefan O'Rear, 21-Mar-2015) (Revised by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses psr1plusg.y Y = PwSer 1 R
psr1plusg.s S = 1 𝑜 mPwSer R
psr1vscafval.n · ˙ = Y
Assertion psr1vsca · ˙ = S

Proof

Step Hyp Ref Expression
1 psr1plusg.y Y = PwSer 1 R
2 psr1plusg.s S = 1 𝑜 mPwSer R
3 psr1vscafval.n · ˙ = Y
4 1 psr1val Y = 1 𝑜 ordPwSer R
5 0ss 1 𝑜 × 1 𝑜
6 5 a1i 1 𝑜 × 1 𝑜
7 2 4 6 opsrvsca S = Y
8 7 mptru S = Y
9 3 8 eqtr4i · ˙ = S