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Metamath Proof Explorer

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 ) ) ) ) ) )