Theorem psrmulval

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

	Ref	Expression
Hypotheses	psrmulr.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	psrmulr.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	psrmulr.m	⊢ · = ( .r ‘ 𝑅 )
	psrmulr.t	⊢ ∙ = ( .r ‘ 𝑆 )
	psrmulr.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
	psrmulfval.i	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
	psrmulfval.r	⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
	psrmulval.r	⊢ ( 𝜑 → 𝑋 ∈ 𝐷 )
Assertion	psrmulval	⊢ ( 𝜑 → ( ( 𝐹 ∙ 𝐺 ) ‘ 𝑋 ) = ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		psrmulr.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2		psrmulr.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
3		psrmulr.m	⊢ · = ( .r ‘ 𝑅 )
4		psrmulr.t	⊢ ∙ = ( .r ‘ 𝑆 )
5		psrmulr.d	⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
6		psrmulfval.i	⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
7		psrmulfval.r	⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
8		psrmulval.r	⊢ ( 𝜑 → 𝑋 ∈ 𝐷 )
9	1 2 3 4 5 6 7	psrmulfval	⊢ ( 𝜑 → ( 𝐹 ∙ 𝐺 ) = ( 𝑥 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) ) ) ) )
10	9	fveq1d	⊢ ( 𝜑 → ( ( 𝐹 ∙ 𝐺 ) ‘ 𝑋 ) = ( ( 𝑥 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) ) ) ) ‘ 𝑋 ) )
11		breq2	⊢ ( 𝑥 = 𝑋 → ( 𝑦 ∘r ≤ 𝑥 ↔ 𝑦 ∘r ≤ 𝑋 ) )
12	11	rabbidv	⊢ ( 𝑥 = 𝑋 → { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } = { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋 } )
13		fvoveq1	⊢ ( 𝑥 = 𝑋 → ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) = ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) )
14	13	oveq2d	⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) = ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) ) )
15	12 14	mpteq12dv	⊢ ( 𝑥 = 𝑋 → ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) ) = ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) ) ) )
16	15	oveq2d	⊢ ( 𝑥 = 𝑋 → ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) ) ) = ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) ) ) ) )
17		eqid	⊢ ( 𝑥 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) ) ) ) = ( 𝑥 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) ) ) )
18		ovex	⊢ ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) ) ) ) ∈ V
19	16 17 18	fvmpt	⊢ ( 𝑋 ∈ 𝐷 → ( ( 𝑥 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) ) ) ) ‘ 𝑋 ) = ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) ) ) ) )
20	8 19	syl	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑥 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑥 ∘f − 𝑘 ) ) ) ) ) ) ‘ 𝑋 ) = ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) ) ) ) )
21	10 20	eqtrd	⊢ ( 𝜑 → ( ( 𝐹 ∙ 𝐺 ) ‘ 𝑋 ) = ( 𝑅 Σg ( 𝑘 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑋 } ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝐺 ‘ ( 𝑋 ∘f − 𝑘 ) ) ) ) ) )