Theorem psrmulfval

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 )
Assertion	psrmulfval	⊢ ( 𝜑 → ( 𝐹 ∙ 𝐺 ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )

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		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
9		fveq1	⊢ ( 𝑔 = 𝐺 → ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) = ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) )
10	8 9	oveqan12d	⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) )
11	10	mpteq2dv	⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) )
12	11	oveq2d	⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) )
13	12	mpteq2dv	⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
14	1 2 3 4 5	psrmulr	⊢ ∙ = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
15		ovex	⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
16	5 15	rabex2	⊢ 𝐷 ∈ V
17	16	mptex	⊢ ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ∈ V
18	13 14 17	ovmpoa	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 ∙ 𝐺 ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
19	6 7 18	syl2anc	⊢ ( 𝜑 → ( 𝐹 ∙ 𝐺 ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )