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Metamath Proof Explorer

Theorem mplcoe4

Description: Decompose a polynomial into a finite sum of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015)

Ref Expression
Hypotheses mplcoe4.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
mplcoe4.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
mplcoe4.z ⊢ 0 = ( 0g ‘ 𝑅 )
mplcoe4.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
mplcoe4.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
mplcoe4.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
mplcoe4.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion mplcoe4 ( 𝜑 → 𝑋 = ( 𝑃 Σg ( 𝑘 ∈ 𝐷 ↦ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 𝑋 ‘ 𝑘 ) , 0 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 mplcoe4.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
2 mplcoe4.d ⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
3 mplcoe4.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 mplcoe4.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
5 mplcoe4.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
6 mplcoe4.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
7 mplcoe4.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
9 eqid ⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ 𝑃 )
10 1 2 3 8 5 4 9 6 7 mplcoe1 ⊢ ( 𝜑 → 𝑋 = ( 𝑃 Σg ( 𝑘 ∈ 𝐷 ↦ ( ( 𝑋 ‘ 𝑘 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 1r ‘ 𝑅 ) , 0 ) ) ) ) ) )
11 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
12 5 adantr ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝐼 ∈ 𝑊 )
13 6 adantr ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑅 ∈ Ring )
14 simpr ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → 𝑘 ∈ 𝐷 )
15 1 11 4 2 7 mplelf ⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) )
16 15 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( 𝑋 ‘ 𝑘 ) ∈ ( Base ‘ 𝑅 ) )
17 1 9 2 8 3 11 12 13 14 16 mplmon2 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐷 ) → ( ( 𝑋 ‘ 𝑘 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 1r ‘ 𝑅 ) , 0 ) ) ) = ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 𝑋 ‘ 𝑘 ) , 0 ) ) )
18 17 mpteq2dva ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐷 ↦ ( ( 𝑋 ‘ 𝑘 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 1r ‘ 𝑅 ) , 0 ) ) ) ) = ( 𝑘 ∈ 𝐷 ↦ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 𝑋 ‘ 𝑘 ) , 0 ) ) ) )
19 18 oveq2d ⊢ ( 𝜑 → ( 𝑃 Σg ( 𝑘 ∈ 𝐷 ↦ ( ( 𝑋 ‘ 𝑘 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 1r ‘ 𝑅 ) , 0 ) ) ) ) ) = ( 𝑃 Σg ( 𝑘 ∈ 𝐷 ↦ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 𝑋 ‘ 𝑘 ) , 0 ) ) ) ) )
20 10 19 eqtrd ⊢ ( 𝜑 → 𝑋 = ( 𝑃 Σg ( 𝑘 ∈ 𝐷 ↦ ( 𝑦 ∈ 𝐷 ↦ if ( 𝑦 = 𝑘 , ( 𝑋 ‘ 𝑘 ) , 0 ) ) ) ) )