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Metamath Proof Explorer

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 ) ) ) ) ) ) )