Theorem psrvsca

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 )
Assertion	psrvsca	⊢ ( 𝜑 → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )

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		sneq	⊢ ( 𝑥 = 𝑋 → { 𝑥 } = { 𝑋 } )
10	9	xpeq2d	⊢ ( 𝑥 = 𝑋 → ( 𝐷 × { 𝑥 } ) = ( 𝐷 × { 𝑋 } ) )
11	10	oveq1d	⊢ ( 𝑥 = 𝑋 → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝑓 ) )
12		oveq2	⊢ ( 𝑓 = 𝐹 → ( ( 𝐷 × { 𝑋 } ) ∘f · 𝑓 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )
13	1 2 3 4 5 6	psrvscafval	⊢ ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) )
14		ovex	⊢ ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ∈ V
15	11 12 13 14	ovmpo	⊢ ( ( 𝑋 ∈ 𝐾 ∧ 𝐹 ∈ 𝐵 ) → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )
16	7 8 15	syl2anc	⊢ ( 𝜑 → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) )