Theorem mplcoe3 21439

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 4490	. . . . . . . . . . 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 2792	. . . . . . . . . 10 ⊢ (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
5	4	mpteq2dv 5207	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ 0))
6		fconstmpt 5694	. . . . . . . . 9 ⊢ (𝐼 × {0}) = (𝑘 ∈ 𝐼 ↦ 0)
7	5, 6	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
8	7	eqeq2d 2747	. . . . . . 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 5207	. . . . 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 7364	. . . . 5 ⊢ (𝑥 = 0 → (𝑥 ↑ (𝑉‘𝑋)) = (0 ↑ (𝑉‘𝑋)))
12	10, 11	eqeq12d 2752	. . . 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 340	. . 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 4490	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
15	14	mpteq2dv 5207	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
16	15	eqeq2d 2747	. . . . . . 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 5207	. . . . 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 7364	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ↑ (𝑉‘𝑋)) = (𝑛 ↑ (𝑉‘𝑋)))
20	18, 19	eqeq12d 2752	. . . 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 340	. . 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 4490	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
23	22	mpteq2dv 5207	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
24	23	eqeq2d 2747	. . . . . . 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 5207	. . . . 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 7364	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ↑ (𝑉‘𝑋)) = ((𝑛 + 1) ↑ (𝑉‘𝑋)))
28	26, 27	eqeq12d 2752	. . . 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 340	. . 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 4490	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
31	30	mpteq2dv 5207	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
32	31	eqeq2d 2747	. . . . . . 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 5207	. . . . 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 7364	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ (𝑉‘𝑋)) = (𝑁 ↑ (𝑉‘𝑋)))
36	34, 35	eqeq12d 2752	. . . 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 340	. . 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 2736	. . . . . 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 21421	. . . . 5 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘𝑃))
45		mplcoe2.g	. . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑃)
46	45, 40	mgpbas 19902	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝐺)
47		eqid 2736	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
48	45, 47	ringidval 19915	. . . . . 6 ⊢ (1r‘𝑃) = (0g‘𝐺)
49		mplcoe2.m	. . . . . 6 ⊢ ↑ = (.g‘𝐺)
50	46, 48, 49	mulg0 18879	. . . . 5 ⊢ ((𝑉‘𝑋) ∈ (Base‘𝑃) → (0 ↑ (𝑉‘𝑋)) = (1r‘𝑃))
51	44, 50	syl 17	. . . 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 21416	. . . 4 ⊢ (𝜑 → (1r‘𝑃) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
56	51, 55	eqtr2d 2777	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 ↑ (𝑉‘𝑋)))
57		oveq1 7364	. . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐼 ∈ 𝑊)
59	42	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
60	52	snifpsrbag 21324	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
61	41, 60	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62		eqid 2736	. . . . . . . . 9 ⊢ (.r‘𝑃) = (.r‘𝑃)
63		1nn0 12429	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
64	63	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
65	52	snifpsrbag 21324	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 1 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
66	41, 64, 65	syl2an 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
67	38, 40, 53, 54, 52, 58, 59, 61, 62, 66	mplmonmul 21437	. . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑋 ∈ 𝐼)
69	39, 52, 53, 54, 58, 59, 68	mvrval 21390	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑉‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
70	69	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉‘𝑋))
71	70	oveq2d 7373	. . . . . . . 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 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ 𝐼) → 𝑛 ∈ ℕ0)
73		0nn0 12428	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
74		ifcl 4531	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
75	72, 73, 74	sylancl 586	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79		eqidd 2737	. . . . . . . . . . . . 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 7637	. . . . . . . . . . . 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 4492	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82		iftrue 4492	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
83	81, 82	oveq12d 7375	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84		iftrue 4492	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
85	83, 84	eqtr4d 2779	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86		00id 11330	. . . . . . . . . . . . . . 15 ⊢ (0 + 0) = 0
87		iffalse 4495	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88		iffalse 4495	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
89	87, 88	oveq12d 7375	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90		iffalse 4495	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
91	86, 89, 90	3eqtr4a 2802	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
92	85, 91	pm2.61i 182	. . . . . . . . . . . . 13 ⊢ (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0)
93	92	mpteq2i 5210	. . . . . . . . . . . 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 2792	. . . . . . . . . . 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 2747	. . . . . . . . . 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 5207	. . . . . . . 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 2785	. . . . . . 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	mplring 21424	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
100	41, 42, 99	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Ring)
101	45	ringmgp 19970	. . . . . . . . . 10 ⊢ (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
102	100, 101	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Mnd)
103	102	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
104		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
105	44	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑉‘𝑋) ∈ (Base‘𝑃))
106	45, 62	mgpplusg 19900	. . . . . . . . 9 ⊢ (.r‘𝑃) = (+g‘𝐺)
107	46, 49, 106	mulgnn0p1 18887	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ (𝑉‘𝑋) ∈ (Base‘𝑃)) → ((𝑛 + 1) ↑ (𝑉‘𝑋)) = ((𝑛 ↑ (𝑉‘𝑋))(.r‘𝑃)(𝑉‘𝑋)))
108	103, 104, 105, 107	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) ↑ (𝑉‘𝑋)) = ((𝑛 ↑ (𝑉‘𝑋))(.r‘𝑃)(𝑉‘𝑋)))
109	98, 108	eqeq12d 2752	. . . . . 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))))
110	57, 109	syl5ibr 245	. . . . 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))))
111	110	expcom 414	. . . 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)))))
112	111	a2d 29	. . 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)))))
113	13, 21, 29, 37, 56, 112	nn0ind 12598	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋))))
114	1, 113	mpcom 38	1 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3407  ifcif 4486  {csn 4586   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632   “ cima 5636  ‘cfv 6496  (class class class)co 7357   ∘_f cof 7615   ↑_m cmap 8765  Fincfn 8883  0cc0 11051  1c1 11052   + caddc 11054  ℕcn 12153  ℕ₀cn0 12413  Basecbs 17083  ._rcmulr 17134  0_gc0g 17321  Mndcmnd 18556  ._gcmg 18872  mulGrpcmgp 19896  1_rcur 19913  Ringcrg 19964   mVar cmvr 21307   mPoly cmpl 21308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-psr 21311  df-mvr 21312  df-mpl 21313

This theorem is referenced by:  mplcoe5  21441  coe1tm  21644