Theorem mplcoe3 22041

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 4527	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 0, 0))
3		ifid 4563	. . . . . . . . . . 11 ⊢ if(𝑘 = 𝑋, 0, 0) = 0
4	2, 3	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
5	4	mpteq2dv 5247	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ 0))
6		fconstmpt 5736	. . . . . . . . 9 ⊢ (𝐼 × {0}) = (𝑘 ∈ 𝐼 ↦ 0)
7	5, 6	eqtr4di 2784	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
8	7	eqeq2d 2737	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝐼 × {0})))
9	8	ifbid 4546	. . . . . 6 ⊢ (𝑥 = 0 → if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
10	9	mpteq2dv 5247	. . . . 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 7423	. . . . 5 ⊢ (𝑥 = 0 → (𝑥 ↑ (𝑉‘𝑋)) = (0 ↑ (𝑉‘𝑋)))
12	10, 11	eqeq12d 2742	. . . 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 339	. . 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 4527	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
15	14	mpteq2dv 5247	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
16	15	eqeq2d 2737	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0))))
17	16	ifbid 4546	. . . . . 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 5247	. . . . 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 7423	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ↑ (𝑉‘𝑋)) = (𝑛 ↑ (𝑉‘𝑋)))
20	18, 19	eqeq12d 2742	. . . 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 339	. . 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 4527	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
23	22	mpteq2dv 5247	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
24	23	eqeq2d 2737	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
25	24	ifbid 4546	. . . . . 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 5247	. . . . 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 7423	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ↑ (𝑉‘𝑋)) = ((𝑛 + 1) ↑ (𝑉‘𝑋)))
28	26, 27	eqeq12d 2742	. . . 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 339	. . 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 4527	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
31	30	mpteq2dv 5247	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
32	31	eqeq2d 2737	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0))))
33	32	ifbid 4546	. . . . . 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 5247	. . . . 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 7423	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ (𝑉‘𝑋)) = (𝑁 ↑ (𝑉‘𝑋)))
36	34, 35	eqeq12d 2742	. . . 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 339	. . 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 2726	. . . . . 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 21997	. . . . 5 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘𝑃))
45		mplcoe2.g	. . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑃)
46	45, 40	mgpbas 20119	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝐺)
47		eqid 2726	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
48	45, 47	ringidval 20162	. . . . . 6 ⊢ (1r‘𝑃) = (0g‘𝐺)
49		mplcoe2.m	. . . . . 6 ⊢ ↑ = (.g‘𝐺)
50	46, 48, 49	mulg0 19064	. . . . 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 22017	. . . 4 ⊢ (𝜑 → (1r‘𝑃) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
56	51, 55	eqtr2d 2767	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 ↑ (𝑉‘𝑋)))
57		oveq1 7423	. . . . . 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 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐼 ∈ 𝑊)
59	42	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
60	52	snifpsrbag 21915	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
61	41, 60	sylan 578	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62		eqid 2726	. . . . . . . . 9 ⊢ (.r‘𝑃) = (.r‘𝑃)
63		1nn0 12534	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
64	63	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
65	52	snifpsrbag 21915	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 1 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
66	41, 64, 65	syl2an 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
67	38, 40, 53, 54, 52, 58, 59, 61, 62, 66	mplmonmul 22039	. . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑋 ∈ 𝐼)
69	39, 52, 53, 54, 58, 59, 68	mvrval 21987	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑉‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
70	69	eqcomd 2732	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉‘𝑋))
71	70	oveq2d 7432	. . . . . . . 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 12533	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
74		ifcl 4568	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
75	72, 73, 74	sylancl 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ 𝐼) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
76	63, 73	ifcli 4570	. . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79		eqidd 2727	. . . . . . . . . . . . 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 7702	. . . . . . . . . . . 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 4529	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82		iftrue 4529	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
83	81, 82	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84		iftrue 4529	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
85	83, 84	eqtr4d 2769	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86		00id 11430	. . . . . . . . . . . . . . 15 ⊢ (0 + 0) = 0
87		iffalse 4532	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88		iffalse 4532	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
89	87, 88	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90		iffalse 4532	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
91	86, 89, 90	3eqtr4a 2792	. . . . . . . . . . . . . 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 5250	. . . . . . . . . . . 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 2782	. . . . . . . . . . 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 2737	. . . . . . . . . 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 4546	. . . . . . . . 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 5247	. . . . . . . 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 2775	. . . . . . 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 22028	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Ring)
100	45	ringmgp 20218	. . . . . . . . . 10 ⊢ (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
101	99, 100	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Mnd)
102	101	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
103		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
104	44	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑉‘𝑋) ∈ (Base‘𝑃))
105	45, 62	mgpplusg 20117	. . . . . . . . 9 ⊢ (.r‘𝑃) = (+g‘𝐺)
106	46, 49, 105	mulgnn0p1 19075	. . . . . . . 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 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) ↑ (𝑉‘𝑋)) = ((𝑛 ↑ (𝑉‘𝑋))(.r‘𝑃)(𝑉‘𝑋)))
108	98, 107	eqeq12d 2742	. . . . . 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 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))))
110	109	expcom 412	. . . 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 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)))))
112	13, 21, 29, 37, 56, 111	nn0ind 12703	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)), 1 , 0 )) = (𝑁 ↑ (𝑉‘𝑋))))
113	1, 112	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 394   = wceq 1534   ∈ wcel 2099  {crab 3419  ifcif 4523  {csn 4623   ↦ cmpt 5228   × cxp 5672  ^◡ccnv 5673   “ cima 5677  ‘cfv 6546  (class class class)co 7416   ∘_f cof 7680   ↑_m cmap 8847  Fincfn 8966  0cc0 11149  1c1 11150   + caddc 11152  ℕcn 12258  ℕ₀cn0 12518  Basecbs 17208  ._rcmulr 17262  0_gc0g 17449  Mndcmnd 18722  ._gcmg 19057  mulGrpcmgp 20113  1_rcur 20160  Ringcrg 20212   mVar cmvr 21898   mPoly cmpl 21899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-isom 6555  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-ofr 7683  df-om 7869  df-1st 7995  df-2nd 7996  df-supp 8167  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-er 8726  df-map 8849  df-pm 8850  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9399  df-sup 9478  df-oi 9546  df-card 9975  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12519  df-z 12605  df-dec 12724  df-uz 12869  df-fz 13533  df-fzo 13676  df-seq 14016  df-hash 14343  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-mulr 17275  df-sca 17277  df-vsca 17278  df-ip 17279  df-tset 17280  df-ple 17281  df-ds 17283  df-hom 17285  df-cco 17286  df-0g 17451  df-gsum 17452  df-prds 17457  df-pws 17459  df-mre 17594  df-mrc 17595  df-acs 17597  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-mhm 18768  df-submnd 18769  df-grp 18926  df-minusg 18927  df-mulg 19058  df-subg 19113  df-ghm 19203  df-cntz 19307  df-cmn 19776  df-abl 19777  df-mgp 20114  df-rng 20132  df-ur 20161  df-ring 20214  df-subrng 20524  df-subrg 20549  df-psr 21902  df-mvr 21903  df-mpl 21904

This theorem is referenced by:  mplcoe5  22043  coe1tm  22260