Theorem psrvscaval

Description: The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014)

	Ref	Expression
Hypotheses	psrvsca.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	psrvsca.n	⊢ ∙ = ( ·𝑠 ‘ 𝑆 )
	psrvsca.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	psrvsca.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	psrvsca.m	⊢ · = ( .r ‘ 𝑅 )
	psrvsca.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
	psrvsca.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
	psrvsca.y	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
	psrvscaval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐷 )
Assertion	psrvscaval	⊢ ( 𝜑 → ( ( 𝑋 ∙ 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		psrvsca.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2		psrvsca.n	⊢ ∙ = ( ·𝑠 ‘ 𝑆 )
3		psrvsca.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
4		psrvsca.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
5		psrvsca.m	⊢ · = ( .r ‘ 𝑅 )
6		psrvsca.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
7		psrvsca.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐾 )
8		psrvsca.y	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
9		psrvscaval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐷 )
10	1 2 3 4 5 6 7 8	psrvsca	⊢ ( 𝜑 → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )
11	10	fveq1d	⊢ ( 𝜑 → ( ( 𝑋 ∙ 𝐹 ) ‘ 𝑌 ) = ( ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ‘ 𝑌 ) )
12		ovex	⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
13	6 12	rabex2	⊢ 𝐷 ∈ V
14	13	a1i	⊢ ( 𝜑 → 𝐷 ∈ V )
15	1 3 6 4 8	psrelbas	⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ 𝐾 )
16	15	ffnd	⊢ ( 𝜑 → 𝐹 Fn 𝐷 )
17		eqidd	⊢ ( ( 𝜑 ∧ 𝑌 ∈ 𝐷 ) → ( 𝐹 ‘ 𝑌 ) = ( 𝐹 ‘ 𝑌 ) )
18	14 7 16 17	ofc1	⊢ ( ( 𝜑 ∧ 𝑌 ∈ 𝐷 ) → ( ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹 ‘ 𝑌 ) ) )
19	9 18	mpdan	⊢ ( 𝜑 → ( ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹 ‘ 𝑌 ) ) )
20	11 19	eqtrd	⊢ ( 𝜑 → ( ( 𝑋 ∙ 𝐹 ) ‘ 𝑌 ) = ( 𝑋 · ( 𝐹 ‘ 𝑌 ) ) )