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	ffvelrnda	⊢ ( ( 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 ) ) ) ) )