Theorem coe1tm 19464

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 2610	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
8	1, 2, 3, 4, 5, 6, 7	ply1tmcl 19463	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ (Base‘𝑃))
9		eqid 2610	. . . 4 ⊢ (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐷 ↑ 𝑋)))
10		eqid 2610	. . . 4 ⊢ (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))
11	9, 7, 2, 10	coe1fval2 19401	. . 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 6007	. . . . 5 ⊢ (𝑥 ∈ ℕ0 → (1𝑜 × {𝑥}):1𝑜⟶ℕ0)
14		nn0ex 11175	. . . . . 6 ⊢ ℕ0 ∈ V
15		1on 7454	. . . . . . 7 ⊢ 1𝑜 ∈ On
16	15	elexi 3186	. . . . . 6 ⊢ 1𝑜 ∈ V
17	14, 16	elmap 7772	. . . . 5 ⊢ ((1𝑜 × {𝑥}) ∈ (ℕ0 ↑𝑚 1𝑜) ↔ (1𝑜 × {𝑥}):1𝑜⟶ℕ0)
18	13, 17	sylibr 223	. . . 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 2611	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})))
21		eqid 2610	. . . . . . . 8 ⊢ (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
22	5, 7	mgpbas 18318	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑁)
23	22	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘𝑁))
24		eqid 2610	. . . . . . . . . 10 ⊢ (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
25		eqid 2610	. . . . . . . . . . 11 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅)
26	2, 25, 7	ply1bas 19386	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
27	24, 26	mgpbas 18318	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅)))
28	27	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅))))
29		ssv 3588	. . . . . . . . 9 ⊢ (Base‘𝑃) ⊆ V
30	29	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) ⊆ V)
31		ovex 6577	. . . . . . . . 9 ⊢ (𝑥(+g‘𝑁)𝑦) ∈ V
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑁)𝑦) ∈ V)
33		eqid 2610	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘𝑃)
34	5, 33	mgpplusg 18316	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘𝑁)
35		eqid 2610	. . . . . . . . . . . . 13 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
36	2, 35, 33	ply1mulr 19418	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘(1𝑜 mPoly 𝑅))
37	24, 36	mgpplusg 18316	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
38	34, 37	eqtr3i 2634	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
39	38	a1i 11	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (+g‘𝑁) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
40	39	oveqdr 6573	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑁)𝑦) = (𝑥(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑦))
41	6, 21, 23, 28, 30, 32, 40	mulgpropd 17407	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ↑ = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
42	41	3ad2ant1 1075	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ↑ = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
43		eqidd 2611	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐷 = 𝐷)
44	3	vr1val 19383	. . . . . . 7 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
45	44	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
46	42, 43, 45	oveq123d 6570	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐷 ↑ 𝑋) = (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
47	46	oveq2d 6565	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) = (𝐶 · (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅))))
48		psr1baslem 19376	. . . . . 6 ⊢ (ℕ0 ↑𝑚 1𝑜) = {𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑎 “ ℕ) ∈ Fin}
49		coe1tm.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
50		eqid 2610	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
51	15	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 1𝑜 ∈ On)
52		eqid 2610	. . . . . 6 ⊢ (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
53		simp1 1054	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝑅 ∈ Ring)
54		0lt1o 7471	. . . . . . 7 ⊢ ∅ ∈ 1𝑜
55	54	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ∅ ∈ 1𝑜)
56		simp3 1056	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0)
57	35, 48, 49, 50, 51, 24, 21, 52, 53, 55, 56	mplcoe3 19287	. . . . 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)‘∅)))
58	57	oveq2d 6565	. . . 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)‘∅))))
59	2, 35, 4	ply1vsca 19417	. . . . 5 ⊢ · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
60		elsni 4142	. . . . . . . . . . 11 ⊢ (𝑏 ∈ {∅} → 𝑏 = ∅)
61		df1o2 7459	. . . . . . . . . . 11 ⊢ 1𝑜 = {∅}
62	60, 61	eleq2s 2706	. . . . . . . . . 10 ⊢ (𝑏 ∈ 1𝑜 → 𝑏 = ∅)
63	62	iftrued 4044	. . . . . . . . 9 ⊢ (𝑏 ∈ 1𝑜 → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
64	63	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑏 ∈ 1𝑜) → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
65	64	mpteq2dva 4672	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (𝑏 ∈ 1𝑜 ↦ 𝐷))
66		fconstmpt 5085	. . . . . . 7 ⊢ (1𝑜 × {𝐷}) = (𝑏 ∈ 1𝑜 ↦ 𝐷)
67	65, 66	syl6eqr 2662	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (1𝑜 × {𝐷}))
68		fconst6g 6007	. . . . . . . 8 ⊢ (𝐷 ∈ ℕ0 → (1𝑜 × {𝐷}):1𝑜⟶ℕ0)
69	14, 16	elmap 7772	. . . . . . . 8 ⊢ ((1𝑜 × {𝐷}) ∈ (ℕ0 ↑𝑚 1𝑜) ↔ (1𝑜 × {𝐷}):1𝑜⟶ℕ0)
70	68, 69	sylibr 223	. . . . . . 7 ⊢ (𝐷 ∈ ℕ0 → (1𝑜 × {𝐷}) ∈ (ℕ0 ↑𝑚 1𝑜))
71	70	3ad2ant3 1077	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (1𝑜 × {𝐷}) ∈ (ℕ0 ↑𝑚 1𝑜))
72	67, 71	eqeltrd 2688	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ∈ (ℕ0 ↑𝑚 1𝑜))
73		simp2 1055	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐶 ∈ 𝐾)
74	35, 59, 48, 50, 49, 1, 51, 53, 72, 73	mplmon2 19314	. . . 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 )))
75	47, 58, 74	3eqtr2d 2650	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) = (𝑦 ∈ (ℕ0 ↑𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
76		eqeq1 2614	. . . 4 ⊢ (𝑦 = (1𝑜 × {𝑥}) → (𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0))))
77	76	ifbid 4058	. . 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 ))
78	19, 20, 75, 77	fmptco 6303	. 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 )))
79	67	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (1𝑜 × {𝐷}))
80	79	eqeq2d 2620	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1𝑜 × {𝑥}) = (1𝑜 × {𝐷})))
81		fveq1 6102	. . . . . . 7 ⊢ ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) → ((1𝑜 × {𝑥})‘∅) = ((1𝑜 × {𝐷})‘∅))
82		vex 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
83	82	fvconst2 6374	. . . . . . . . 9 ⊢ (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
84	54, 83	mp1i 13	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥})‘∅) = 𝑥)
85		simpl3 1059	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐷 ∈ ℕ0)
86		fvconst2g 6372	. . . . . . . . 9 ⊢ ((𝐷 ∈ ℕ0 ∧ ∅ ∈ 1𝑜) → ((1𝑜 × {𝐷})‘∅) = 𝐷)
87	85, 54, 86	sylancl 693	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝐷})‘∅) = 𝐷)
88	84, 87	eqeq12d 2625	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((1𝑜 × {𝑥})‘∅) = ((1𝑜 × {𝐷})‘∅) ↔ 𝑥 = 𝐷))
89	81, 88	syl5ib 233	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) → 𝑥 = 𝐷))
90		sneq 4135	. . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝑥} = {𝐷})
91	90	xpeq2d 5063	. . . . . 6 ⊢ (𝑥 = 𝐷 → (1𝑜 × {𝑥}) = (1𝑜 × {𝐷}))
92	89, 91	impbid1 214	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) ↔ 𝑥 = 𝐷))
93	80, 92	bitrd 267	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ 𝑥 = 𝐷))
94	93	ifbid 4058	. . 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 ))
95	94	mpteq2dva 4672	. 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 )))
96	12, 78, 95	3eqtrd 2648	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 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440   ↑_𝑚 cmap 7744  0cc0 9815  ℕ₀cn0 11169  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769   ·_𝑠 cvsca 15772  0_gc0g 15923  ._gcmg 17363  mulGrpcmgp 18312  1_rcur 18324  Ringcrg 18370   mVar cmvr 19173   mPoly cmpl 19174  PwSer₁cps1 19366  var₁cv1 19367  Poly₁cpl1 19368  coe₁cco1 19369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-ple 15788  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374

This theorem is referenced by:  coe1tmfv1  19465  coe1tmfv2  19466  coe1scl  19478  gsummoncoe1  19495  decpmatid  20394  monmatcollpw  20403  mp2pm2mplem4  20433