Theorem mplcoe3 22377

Description: Decompose a monomial in one variable into a power of a variable. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 18-Jul-2019.)

Hypotheses

Ref	Expression
mplcoe1.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplcoe1.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
mplcoe1.z	⊢ 0 = (0g‘𝑅)
mplcoe1.o	⊢ 1 = (1r‘𝑅)
mplcoe1.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplcoe2.g	⊢ 𝐺 = (mulGrp‘𝑃)
mplcoe2.m	⊢ ↑ = (.g‘𝐺)
mplcoe2.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
mplcoe3.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mplcoe3.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)
mplcoe3.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Assertion

Ref	Expression
mplcoe3	⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋)))

Distinct variable groups:   ↑ ,𝑘   𝑦,𝑘, 1   𝑘,𝐺   𝑓,𝑘,𝑦,𝐼   𝑘,𝑁,𝑦   𝜑,𝑘,𝑦   𝑅,𝑓,𝑦   𝐷,𝑘,𝑦   𝑃,𝑘   𝑘,𝑉   0 ,𝑓,𝑘,𝑦   𝑓,𝑋,𝑘,𝑦   𝑘,𝑊,𝑦

Allowed substitution hints:   𝜑(𝑓)   𝐷(𝑓)   𝑃(𝑦, 𝑓)   𝑅(𝑘)   1 (𝑓)   ↑ (𝑦, 𝑓)   𝐺(𝑦, 𝑓)   𝑁(𝑓)   𝑉(𝑦, 𝑓)   𝑊(𝑓)

Proof of Theorem mplcoe3

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplcoe3.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		ifeq1 4489	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 0, 0))
3		ifid 4526	. . . . . . . . . . 11 ⊢ if(𝑘 = 𝑋, 0, 0) = 0
4	2, 3	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
5	4	mpteq2dv 5202	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ 0))
6		fconstmpt 5717	. . . . . . . . 9 ⊢ (𝐼 × {0}) = (𝑘 ∈ 𝐼 ↦ 0)
7	5, 6	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
8	7	eqeq2d 2774	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝐼 × {0})))
9	8	ifbid 4509	. . . . . 6 ⊢ (𝑥 = 0 → if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
10	9	mpteq2dv 5202	. . . . 5 ⊢ (𝑥 = 0 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
11		oveq1 7450	. . . . 5 ⊢ (𝑥 = 0 → (𝑥 ↑ (𝑉‘𝑋)) = (0 ↑ (𝑉‘𝑋)))
12	10, 11	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 0 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 ↑ (𝑉‘𝑋)) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 ↑ (𝑉‘𝑋))))
13	12	imbi2d 343	. . 3 ⊢ (𝑥 = 0 → ((𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 ↑ (𝑉‘𝑋))) ↔ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 ↑ (𝑉‘𝑋)))))
14		ifeq1 4489	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
15	14	mpteq2dv 5202	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
16	15	eqeq2d 2774	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0))))
17	16	ifbid 4509	. . . . . 6 ⊢ (𝑥 = 𝑛 → if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))
18	17	mpteq2dv 5202	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )))
19		oveq1 7450	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ↑ (𝑉‘𝑋)) = (𝑛 ↑ (𝑉‘𝑋)))
20	18, 19	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 ↑ (𝑉‘𝑋)) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 ↑ (𝑉‘𝑋))))
21	20	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 ↑ (𝑉‘𝑋))) ↔ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 ↑ (𝑉‘𝑋)))))
22		ifeq1 4489	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
23	22	mpteq2dv 5202	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
24	23	eqeq2d 2774	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
25	24	ifbid 4509	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 ))
26	25	mpteq2dv 5202	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )))
27		oveq1 7450	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ↑ (𝑉‘𝑋)) = ((𝑛 + 1) ↑ (𝑉‘𝑋)))
28	26, 27	eqeq12d 2779	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 ↑ (𝑉‘𝑋)) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) ↑ (𝑉‘𝑋))))
29	28	imbi2d 343	. . 3 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 ↑ (𝑉‘𝑋))) ↔ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) ↑ (𝑉‘𝑋)))))
30		ifeq1 4489	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
31	30	mpteq2dv 5202	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
32	31	eqeq2d 2774	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0))))
33	32	ifbid 4509	. . . . . 6 ⊢ (𝑥 = 𝑁 → if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 ))
34	33	mpteq2dv 5202	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )))
35		oveq1 7450	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ (𝑉‘𝑋)) = (𝑁 ↑ (𝑉‘𝑋)))
36	34, 35	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 ↑ (𝑉‘𝑋)) ↔ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋))))
37	36	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 )) = (𝑥 ↑ (𝑉‘𝑋))) ↔ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋)))))
38		mplcoe1.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
39		mplcoe2.v	. . . . . 6 ⊢ 𝑉 = (𝐼 mVar 𝑅)
40		eqid 2763	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
41		mplcoe1.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
42		mplcoe3.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
43		mplcoe3.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
44	38, 39, 40, 41, 42, 43	mvrcl 22329	. . . . 5 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘𝑃))
45		mplcoe2.g	. . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑃)
46	45, 40	mgpbas 20395	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝐺)
47		eqid 2763	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
48	45, 47	ringidval 20439	. . . . . 6 ⊢ (1r‘𝑃) = (0g‘𝐺)
49		mplcoe2.m	. . . . . 6 ⊢ ↑ = (.g‘𝐺)
50	46, 48, 49	mulg0 19313	. . . . 5 ⊢ ((𝑉‘𝑋) ∈ (Base‘𝑃) → (0 ↑ (𝑉‘𝑋)) = (1r‘𝑃))
51	44, 50	syl 18	. . . 4 ⊢ (𝜑 → (0 ↑ (𝑉‘𝑋)) = (1r‘𝑃))
52		mplcoe1.d	. . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
53		mplcoe1.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
54		mplcoe1.o	. . . . 5 ⊢ 1 = (1r‘𝑅)
55	38, 52, 53, 54, 47, 41, 42	mpl1 22349	. . . 4 ⊢ (𝜑 → (1r‘𝑃) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
56	51, 55	eqtr2d 2799	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 ↑ (𝑉‘𝑋)))
57		oveq1 7450	. . . . . 6 ⊢ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 ↑ (𝑉‘𝑋)) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r‘𝑃)(𝑉‘𝑋)) = ((𝑛 ↑ (𝑉‘𝑋))(.r‘𝑃)(𝑉‘𝑋)))
58	41	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐼 ∈ 𝑊)
59	42	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
60	52	snifpsrbag 22258	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
61	41, 60	sylan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62		eqid 2763	. . . . . . . . 9 ⊢ (.r‘𝑃) = (.r‘𝑃)
63		1nn0 12649	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
64	63	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
65	52	snifpsrbag 22258	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 1 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
66	41, 64, 65	syl2an 608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
67	38, 40, 53, 54, 52, 58, 59, 61, 62, 66	mplmonmul 22375	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0))), 1 , 0 )))
68	43	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑋 ∈ 𝐼)
69	39, 52, 53, 54, 58, 59, 68	mvrval 22319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑉‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
70	69	eqcomd 2769	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉‘𝑋))
71	70	oveq2d 7459	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 ))) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r‘𝑃)(𝑉‘𝑋)))
72		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ 𝐼) → 𝑛 ∈ ℕ0)
73		0nn0 12648	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
74		ifcl 4531	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
75	72, 73, 74	sylancl 598	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ 𝐼) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
76	63, 73	ifcli 4533	. . . . . . . . . . . . . 14 ⊢ if(𝑘 = 𝑋, 1, 0) ∈ ℕ0
77	76	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ 𝐼) → if(𝑘 = 𝑋, 1, 0) ∈ ℕ0)
78		eqidd 2764	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79		eqidd 2764	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)))
80	58, 75, 77, 78, 79	offval2 7737	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) = (𝑘 ∈ 𝐼 ↦ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0))))
81		iftrue 4491	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82		iftrue 4491	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
83	81, 82	oveq12d 7461	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84		iftrue 4491	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
85	83, 84	eqtr4d 2801	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86		00id 11512	. . . . . . . . . . . . . . 15 ⊢ (0 + 0) = 0
87		iffalse 4494	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88		iffalse 4494	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
89	87, 88	oveq12d 7461	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90		iffalse 4494	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
91	86, 89, 90	3eqtr4a 2824	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
92	85, 91	pm2.61i 184	. . . . . . . . . . . . 13 ⊢ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0)
93	92	mpteq2i 5204	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐼 ↦ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0))) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))
94	80, 93	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
95	94	eqeq2d 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑦 = ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0))) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
96	95	ifbid 4509	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → if(𝑦 = ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0))), 1 , 0 ) = if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 ))
97	96	mpteq2dv 5202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∘f + (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0))), 1 , 0 )) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )))
98	67, 71, 97	3eqtr3rd 2807	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r‘𝑃)(𝑉‘𝑋)))
99	38, 41, 42	mplringd 22360	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Ring)
100	45	ringmgp 20495	. . . . . . . . . 10 ⊢ (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
101	99, 100	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Mnd)
102	101	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
103		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
104	44	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑉‘𝑋) ∈ (Base‘𝑃))
105	45, 62	mgpplusg 20394	. . . . . . . . 9 ⊢ (.r‘𝑃) = (+g‘𝐺)
106	46, 49, 105	mulgnn0p1 19324	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ (𝑉‘𝑋) ∈ (Base‘𝑃)) → ((𝑛 + 1) ↑ (𝑉‘𝑋)) = ((𝑛 ↑ (𝑉‘𝑋))(.r‘𝑃)(𝑉‘𝑋)))
107	102, 103, 104, 106	syl3anc 1399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) ↑ (𝑉‘𝑋)) = ((𝑛 ↑ (𝑉‘𝑋))(.r‘𝑃)(𝑉‘𝑋)))
108	98, 107	eqeq12d 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) ↑ (𝑉‘𝑋)) ↔ ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 ))(.r‘𝑃)(𝑉‘𝑋)) = ((𝑛 ↑ (𝑉‘𝑋))(.r‘𝑃)(𝑉‘𝑋))))
109	57, 108	imbitrrid 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 ↑ (𝑉‘𝑋)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) ↑ (𝑉‘𝑋))))
110	109	expcom 419	. . . 4 ⊢ (𝑛 ∈ ℕ0 → (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 ↑ (𝑉‘𝑋)) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) ↑ (𝑉‘𝑋)))))
111	110	a2d 30	. . 3 ⊢ (𝑛 ∈ ℕ0 → ((𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)), 1 , 0 )) = (𝑛 ↑ (𝑉‘𝑋))) → (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)), 1 , 0 )) = ((𝑛 + 1) ↑ (𝑉‘𝑋)))))
112	13, 21, 29, 37, 56, 111	nn0ind 12821	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋))))
113	1, 112	mpcom 39	1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  ifcif 4485  {csn 4588   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654   “ cima 5658  ‘cfv 6558  (class class class)co 7443   ∘_f cof 7715   ↑_m cmap 8875  Fincfn 9001  0cc0 11227  1c1 11228   + caddc 11230  ℕcn 12362  ℕ₀cn0 12633  Basecbs 17416  ._rcmulr 17458  0_gc0g 17639  Mndcmnd 18952  ._gcmg 19306  mulGrpcmgp 20390  1_rcur 20437  Ringcrg 20489   mVar cmvr 22243   mPoly cmpl 22244

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-ofr 7718  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-sup 9461  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-fzo 13816  df-seq 14172  df-hash 14502  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-hom 17481  df-cco 17482  df-0g 17641  df-gsum 17642  df-prds 17647  df-pws 17649  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-mulg 19307  df-subg 19362  df-ghm 19457  df-cntz 19561  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-subrng 20828  df-subrg 20852  df-psr 22247  df-mvr 22248  df-mpl 22249

This theorem is used by:  mplcoe5  22379  coe1tm  22622