Theorem mplcoe3 22025

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 4471	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 0, 0))
3		ifid 4508	. . . . . . . . . . 11 ⊢ if(𝑘 = 𝑋, 0, 0) = 0
4	2, 3	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥 = 0 → if(𝑘 = 𝑋, 𝑥, 0) = 0)
5	4	mpteq2dv 5180	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ 0))
6		fconstmpt 5684	. . . . . . . . 9 ⊢ (𝐼 × {0}) = (𝑘 ∈ 𝐼 ↦ 0)
7	5, 6	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝐼 × {0}))
8	7	eqeq2d 2748	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝐼 × {0})))
9	8	ifbid 4491	. . . . . 6 ⊢ (𝑥 = 0 → if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)), 1 , 0 ) = if(𝑦 = (𝐼 × {0}), 1 , 0 ))
10	9	mpteq2dv 5180	. . . . 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 7365	. . . . 5 ⊢ (𝑥 = 0 → (𝑥 ↑ (𝑉‘𝑋)) = (0 ↑ (𝑉‘𝑋)))
12	10, 11	eqeq12d 2753	. . . 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 4471	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑛, 0))
15	14	mpteq2dv 5180	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
16	15	eqeq2d 2748	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0))))
17	16	ifbid 4491	. . . . . 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 5180	. . . . 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 7365	. . . . 5 ⊢ (𝑥 = 𝑛 → (𝑥 ↑ (𝑉‘𝑋)) = (𝑛 ↑ (𝑉‘𝑋)))
20	18, 19	eqeq12d 2753	. . . 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 4471	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
23	22	mpteq2dv 5180	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0)))
24	23	eqeq2d 2748	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, (𝑛 + 1), 0))))
25	24	ifbid 4491	. . . . . 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 5180	. . . . 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 7365	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ↑ (𝑉‘𝑋)) = ((𝑛 + 1) ↑ (𝑉‘𝑋)))
28	26, 27	eqeq12d 2753	. . . 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 4471	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → if(𝑘 = 𝑋, 𝑥, 0) = if(𝑘 = 𝑋, 𝑁, 0))
31	30	mpteq2dv 5180	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0)))
32	31	eqeq2d 2748	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑥, 0)) ↔ 𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑁, 0))))
33	32	ifbid 4491	. . . . . 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 5180	. . . . 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 7365	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ (𝑉‘𝑋)) = (𝑁 ↑ (𝑉‘𝑋)))
36	34, 35	eqeq12d 2753	. . . 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 2737	. . . . . 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 21979	. . . . 5 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘𝑃))
45		mplcoe2.g	. . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑃)
46	45, 40	mgpbas 20115	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝐺)
47		eqid 2737	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
48	45, 47	ringidval 20153	. . . . . 6 ⊢ (1r‘𝑃) = (0g‘𝐺)
49		mplcoe2.m	. . . . . 6 ⊢ ↑ = (.g‘𝐺)
50	46, 48, 49	mulg0 19039	. . . . 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 21999	. . . 4 ⊢ (𝜑 → (1r‘𝑃) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )))
56	51, 55	eqtr2d 2773	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 1 , 0 )) = (0 ↑ (𝑉‘𝑋)))
57		oveq1 7365	. . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐼 ∈ 𝑊)
59	42	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
60	52	snifpsrbag 21908	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
61	41, 60	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) ∈ 𝐷)
62		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝑃) = (.r‘𝑃)
63		1nn0 12442	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
64	63	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → 1 ∈ ℕ0)
65	52	snifpsrbag 21908	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 1 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
66	41, 64, 65	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)) ∈ 𝐷)
67	38, 40, 53, 54, 52, 58, 59, 61, 62, 66	mplmonmul 22023	. . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑋 ∈ 𝐼)
69	39, 52, 53, 54, 58, 59, 68	mvrval 21969	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑉‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )))
70	69	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 1, 0)), 1 , 0 )) = (𝑉‘𝑋))
71	70	oveq2d 7374	. . . . . . . 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 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ 𝐼) → 𝑛 ∈ ℕ0)
73		0nn0 12441	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
74		ifcl 4513	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0 ∧ 0 ∈ ℕ0) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
75	72, 73, 74	sylancl 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ 𝐼) → if(𝑘 = 𝑋, 𝑛, 0) ∈ ℕ0)
76	63, 73	ifcli 4515	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑋, 𝑛, 0)))
79		eqidd 2738	. . . . . . . . . . . . 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 7642	. . . . . . . . . . . 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 4473	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 𝑛)
82		iftrue 4473	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 1)
83	81, 82	oveq12d 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (𝑛 + 1))
84		iftrue 4473	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = (𝑛 + 1))
85	83, 84	eqtr4d 2775	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = if(𝑘 = 𝑋, (𝑛 + 1), 0))
86		00id 11310	. . . . . . . . . . . . . . 15 ⊢ (0 + 0) = 0
87		iffalse 4476	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 𝑛, 0) = 0)
88		iffalse 4476	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, 1, 0) = 0)
89	87, 88	oveq12d 7376	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑋 → (if(𝑘 = 𝑋, 𝑛, 0) + if(𝑘 = 𝑋, 1, 0)) = (0 + 0))
90		iffalse 4476	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑋 → if(𝑘 = 𝑋, (𝑛 + 1), 0) = 0)
91	86, 89, 90	3eqtr4a 2798	. . . . . . . . . . . . . 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 5182	. . . . . . . . . . . 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 2788	. . . . . . . . . . 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 2748	. . . . . . . . . 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 4491	. . . . . . . . 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 5180	. . . . . . . 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 2781	. . . . . . 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 22010	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Ring)
100	45	ringmgp 20209	. . . . . . . . . 10 ⊢ (𝑃 ∈ Ring → 𝐺 ∈ Mnd)
101	99, 100	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ Mnd)
102	101	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐺 ∈ Mnd)
103		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
104	44	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑉‘𝑋) ∈ (Base‘𝑃))
105	45, 62	mgpplusg 20114	. . . . . . . . 9 ⊢ (.r‘𝑃) = (+g‘𝐺)
106	46, 49, 105	mulgnn0p1 19050	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) ↑ (𝑉‘𝑋)) = ((𝑛 ↑ (𝑉‘𝑋))(.r‘𝑃)(𝑉‘𝑋)))
108	98, 107	eqeq12d 2753	. . . . . 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 246	. . . . 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 413	. . . 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 12613	. 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 395   = wceq 1542   ∈ wcel 2114  {crab 3390  ifcif 4467  {csn 4568   ↦ cmpt 5167   × cxp 5620  ^◡ccnv 5621   “ cima 5625  ‘cfv 6490  (class class class)co 7358   ∘_f cof 7620   ↑_m cmap 8764  Fincfn 8884  0cc0 11027  1c1 11028   + caddc 11030  ℕcn 12163  ℕ₀cn0 12426  Basecbs 17168  ._rcmulr 17210  0_gc0g 17391  Mndcmnd 18691  ._gcmg 19032  mulGrpcmgp 20110  1_rcur 20151  Ringcrg 20203   mVar cmvr 21893   mPoly cmpl 21894

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8102  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-2o 8397  df-er 8634  df-map 8766  df-pm 8767  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9266  df-sup 9346  df-oi 9416  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-7 12238  df-8 12239  df-9 12240  df-n0 12427  df-z 12514  df-dec 12634  df-uz 12778  df-fz 13451  df-fzo 13598  df-seq 13953  df-hash 14282  df-struct 17106  df-sets 17123  df-slot 17141  df-ndx 17153  df-base 17169  df-ress 17190  df-plusg 17222  df-mulr 17223  df-sca 17225  df-vsca 17226  df-ip 17227  df-tset 17228  df-ple 17229  df-ds 17231  df-hom 17233  df-cco 17234  df-0g 17393  df-gsum 17394  df-prds 17399  df-pws 17401  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-mhm 18740  df-submnd 18741  df-grp 18901  df-minusg 18902  df-mulg 19033  df-subg 19088  df-ghm 19177  df-cntz 19281  df-cmn 19746  df-abl 19747  df-mgp 20111  df-rng 20123  df-ur 20152  df-ring 20205  df-subrng 20512  df-subrg 20536  df-psr 21897  df-mvr 21898  df-mpl 21899

This theorem is referenced by:  mplcoe5  22027  coe1tm  22247