Theorem coe1tm 19691

Description: Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Hypotheses

Ref	Expression
coe1tm.z	⊢ 0 = (0g‘𝑅)
coe1tm.k	⊢ 𝐾 = (Base‘𝑅)
coe1tm.p	⊢ 𝑃 = (Poly1‘𝑅)
coe1tm.x	⊢ 𝑋 = (var1‘𝑅)
coe1tm.m	⊢ · = ( ·𝑠 ‘𝑃)
coe1tm.n	⊢ 𝑁 = (mulGrp‘𝑃)
coe1tm.e	⊢ ↑ = (.g‘𝑁)

Assertion

Ref	Expression
coe1tm	⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))

Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥, ↑   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝑥,𝑅   𝑥, ·

Proof of Theorem coe1tm

Dummy variables 𝑎 𝑏 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1tm.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
2		coe1tm.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
3		coe1tm.x	. . . 4 ⊢ 𝑋 = (var1‘𝑅)
4		coe1tm.m	. . . 4 ⊢ · = ( ·𝑠 ‘𝑃)
5		coe1tm.n	. . . 4 ⊢ 𝑁 = (mulGrp‘𝑃)
6		coe1tm.e	. . . 4 ⊢ ↑ = (.g‘𝑁)
7		eqid 2651	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
8	1, 2, 3, 4, 5, 6, 7	ply1tmcl 19690	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ (Base‘𝑃))
9		eqid 2651	. . . 4 ⊢ (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐷 ↑ 𝑋)))
10		eqid 2651	. . . 4 ⊢ (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))
11	9, 7, 2, 10	coe1fval2 19628	. . 3 ⊢ ((𝐶 · (𝐷 ↑ 𝑋)) ∈ (Base‘𝑃) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = ((𝐶 · (𝐷 ↑ 𝑋)) ∘ (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))))
12	8, 11	syl 17	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = ((𝐶 · (𝐷 ↑ 𝑋)) ∘ (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))))
13		fconst6g 6132	. . . . 5 ⊢ (𝑥 ∈ ℕ0 → (1𝑜 × {𝑥}):1𝑜⟶ℕ0)
14		nn0ex 11336	. . . . . 6 ⊢ ℕ0 ∈ V
15		1on 7612	. . . . . . 7 ⊢ 1𝑜 ∈ On
16	15	elexi 3244	. . . . . 6 ⊢ 1𝑜 ∈ V
17	14, 16	elmap 7928	. . . . 5 ⊢ ((1𝑜 × {𝑥}) ∈ (ℕ0 ↑𝑚 1𝑜) ↔ (1𝑜 × {𝑥}):1𝑜⟶ℕ0)
18	13, 17	sylibr 224	. . . 4 ⊢ (𝑥 ∈ ℕ0 → (1𝑜 × {𝑥}) ∈ (ℕ0 ↑𝑚 1𝑜))
19	18	adantl 481	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (1𝑜 × {𝑥}) ∈ (ℕ0 ↑𝑚 1𝑜))
20		eqidd 2652	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})))
21		eqid 2651	. . . . . . . 8 ⊢ (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
22	5, 7	mgpbas 18541	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑁)
23	22	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘𝑁))
24		eqid 2651	. . . . . . . . . 10 ⊢ (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
25		eqid 2651	. . . . . . . . . . 11 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
26	2, 25, 7	ply1bas 19613	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
27	24, 26	mgpbas 18541	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅)))
28	27	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅))))
29		ssv 3658	. . . . . . . . 9 ⊢ (Base‘𝑃) ⊆ V
30	29	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) ⊆ V)
31		ovexd 6720	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑁)𝑦) ∈ V)
32		eqid 2651	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘𝑃)
33	5, 32	mgpplusg 18539	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘𝑁)
34		eqid 2651	. . . . . . . . . . . . 13 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
35	2, 34, 32	ply1mulr 19645	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘(1𝑜 mPoly 𝑅))
36	24, 35	mgpplusg 18539	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
37	33, 36	eqtr3i 2675	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
38	37	a1i 11	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (+g‘𝑁) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
39	38	oveqdr 6714	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑁)𝑦) = (𝑥(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑦))
40	6, 21, 23, 28, 30, 31, 39	mulgpropd 17631	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ↑ = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
41	40	3ad2ant1 1102	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ↑ = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
42		eqidd 2652	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐷 = 𝐷)
43	3	vr1val 19610	. . . . . . 7 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
44	43	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
45	41, 42, 44	oveq123d 6711	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐷 ↑ 𝑋) = (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
46	45	oveq2d 6706	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) = (𝐶 · (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅))))
47		psr1baslem 19603	. . . . . 6 ⊢ (ℕ0 ↑𝑚 1𝑜) = {𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑎 “ ℕ) ∈ Fin}
48		coe1tm.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
49		eqid 2651	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
50	15	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 1𝑜 ∈ On)
51		eqid 2651	. . . . . 6 ⊢ (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
52		simp1 1081	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝑅 ∈ Ring)
53		0lt1o 7629	. . . . . . 7 ⊢ ∅ ∈ 1𝑜
54	53	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ∅ ∈ 1𝑜)
55		simp3 1083	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0)
56	34, 47, 48, 49, 50, 24, 21, 51, 52, 54, 55	mplcoe3 19514	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), (1r‘𝑅), 0 )) = (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
57	56	oveq2d 6706	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), (1r‘𝑅), 0 ))) = (𝐶 · (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅))))
58	2, 34, 4	ply1vsca 19644	. . . . 5 ⊢ · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
59		elsni 4227	. . . . . . . . . . 11 ⊢ (𝑏 ∈ {∅} → 𝑏 = ∅)
60		df1o2 7617	. . . . . . . . . . 11 ⊢ 1𝑜 = {∅}
61	59, 60	eleq2s 2748	. . . . . . . . . 10 ⊢ (𝑏 ∈ 1𝑜 → 𝑏 = ∅)
62	61	iftrued 4127	. . . . . . . . 9 ⊢ (𝑏 ∈ 1𝑜 → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
63	62	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑏 ∈ 1𝑜) → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
64	63	mpteq2dva 4777	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (𝑏 ∈ 1𝑜 ↦ 𝐷))
65		fconstmpt 5197	. . . . . . 7 ⊢ (1𝑜 × {𝐷}) = (𝑏 ∈ 1𝑜 ↦ 𝐷)
66	64, 65	syl6eqr 2703	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (1𝑜 × {𝐷}))
67		fconst6g 6132	. . . . . . . 8 ⊢ (𝐷 ∈ ℕ0 → (1𝑜 × {𝐷}):1𝑜⟶ℕ0)
68	14, 16	elmap 7928	. . . . . . . 8 ⊢ ((1𝑜 × {𝐷}) ∈ (ℕ0 ↑𝑚 1𝑜) ↔ (1𝑜 × {𝐷}):1𝑜⟶ℕ0)
69	67, 68	sylibr 224	. . . . . . 7 ⊢ (𝐷 ∈ ℕ0 → (1𝑜 × {𝐷}) ∈ (ℕ0 ↑𝑚 1𝑜))
70	69	3ad2ant3 1104	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (1𝑜 × {𝐷}) ∈ (ℕ0 ↑𝑚 1𝑜))
71	66, 70	eqeltrd 2730	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ∈ (ℕ0 ↑𝑚 1𝑜))
72		simp2 1082	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐶 ∈ 𝐾)
73	34, 58, 47, 49, 48, 1, 50, 52, 71, 72	mplmon2 19541	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), (1r‘𝑅), 0 ))) = (𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
74	46, 57, 73	3eqtr2d 2691	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) = (𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
75		eqeq1 2655	. . . 4 ⊢ (𝑦 = (1𝑜 × {𝑥}) → (𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0))))
76	75	ifbid 4141	. . 3 ⊢ (𝑦 = (1𝑜 × {𝑥}) → if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ) = if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ))
77	19, 20, 74, 76	fmptco 6436	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ((𝐶 · (𝐷 ↑ 𝑋)) ∘ (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))) = (𝑥 ∈ ℕ0 ↦ if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
78	66	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (1𝑜 × {𝐷}))
79	78	eqeq2d 2661	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1𝑜 × {𝑥}) = (1𝑜 × {𝐷})))
80		fveq1 6228	. . . . . . 7 ⊢ ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) → ((1𝑜 × {𝑥})‘∅) = ((1𝑜 × {𝐷})‘∅))
81		vex 3234	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
82	81	fvconst2 6510	. . . . . . . . 9 ⊢ (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
83	53, 82	mp1i 13	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥})‘∅) = 𝑥)
84		simpl3 1086	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐷 ∈ ℕ0)
85		fvconst2g 6508	. . . . . . . . 9 ⊢ ((𝐷 ∈ ℕ0 ∧ ∅ ∈ 1𝑜) → ((1𝑜 × {𝐷})‘∅) = 𝐷)
86	84, 53, 85	sylancl 695	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝐷})‘∅) = 𝐷)
87	83, 86	eqeq12d 2666	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((1𝑜 × {𝑥})‘∅) = ((1𝑜 × {𝐷})‘∅) ↔ 𝑥 = 𝐷))
88	80, 87	syl5ib 234	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) → 𝑥 = 𝐷))
89		sneq 4220	. . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝑥} = {𝐷})
90	89	xpeq2d 5173	. . . . . 6 ⊢ (𝑥 = 𝐷 → (1𝑜 × {𝑥}) = (1𝑜 × {𝐷}))
91	88, 90	impbid1 215	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) ↔ 𝑥 = 𝐷))
92	79, 91	bitrd 268	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ 𝑥 = 𝐷))
93	92	ifbid 4141	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ) = if(𝑥 = 𝐷, 𝐶, 0 ))
94	93	mpteq2dva 4777	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))
95	12, 77, 94	3eqtrd 2689	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  ifcif 4119  {csn 4210   ↦ cmpt 4762   × cxp 5141   ∘ ccom 5147  Oncon0 5761  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  1_𝑜c1o 7598   ↑_𝑚 cmap 7899  0cc0 9974  ℕ₀cn0 11330  Basecbs 15904  +_gcplusg 15988  ._rcmulr 15989   ·_𝑠 cvsca 15992  0_gc0g 16147  ._gcmg 17587  mulGrpcmgp 18535  1_rcur 18547  Ringcrg 18593   mVar cmvr 19400   mPoly cmpl 19401  PwSer₁cps1 19593  var₁cv1 19594  Poly₁cpl1 19595  coe₁cco1 19596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-tset 16007  df-ple 16008  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601

This theorem is referenced by:  coe1tmfv1  19692  coe1tmfv2  19693  coe1scl  19705  gsummoncoe1  19722  decpmatid  20623  monmatcollpw  20632  mp2pm2mplem4  20662