Theorem coe1tm 22402

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 2769	. . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃)
8	1, 2, 3, 4, 5, 6, 7	ply1tmcl 22401	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ (Base‘𝑃))
9		eqid 2769	. . . 4 ⊢ (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐷 ↑ 𝑋)))
10		eqid 2769	. . . 4 ⊢ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))
11	9, 7, 2, 10	coe1fval2 22338	. . 3 ⊢ ((𝐶 · (𝐷 ↑ 𝑋)) ∈ (Base‘𝑃) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = ((𝐶 · (𝐷 ↑ 𝑋)) ∘ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))))
12	8, 11	syl 18	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = ((𝐶 · (𝐷 ↑ 𝑋)) ∘ (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))))
13		fconst6g 6768	. . . . 5 ⊢ (𝑥 ∈ ℕ0 → (1o × {𝑥}):1o⟶ℕ0)
14		nn0ex 12509	. . . . . 6 ⊢ ℕ0 ∈ V
15		1oex 8462	. . . . . 6 ⊢ 1o ∈ V
16	14, 15	elmap 8868	. . . . 5 ⊢ ((1o × {𝑥}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {𝑥}):1o⟶ℕ0)
17	13, 16	sylibr 237	. . . 4 ⊢ (𝑥 ∈ ℕ0 → (1o × {𝑥}) ∈ (ℕ0 ↑m 1o))
18	17	adantl 486	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (1o × {𝑥}) ∈ (ℕ0 ↑m 1o))
19		eqidd 2770	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})))
20		eqid 2769	. . . . . . . 8 ⊢ (.g‘(mulGrp‘(1o mPoly 𝑅))) = (.g‘(mulGrp‘(1o mPoly 𝑅)))
21	5, 7	mgpbas 20220	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘𝑁)
22	21	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘𝑁))
23		eqid 2769	. . . . . . . . . 10 ⊢ (mulGrp‘(1o mPoly 𝑅)) = (mulGrp‘(1o mPoly 𝑅))
24	2, 7	ply1bas 22323	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅))
25	23, 24	mgpbas 20220	. . . . . . . . 9 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘(1o mPoly 𝑅)))
26	25	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘(mulGrp‘(1o mPoly 𝑅))))
27		ssv 3969	. . . . . . . . 9 ⊢ (Base‘𝑃) ⊆ V
28	27	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) ⊆ V)
29		ovexd 7446	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑁)𝑦) ∈ V)
30		eqid 2769	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘𝑃)
31	5, 30	mgpplusg 20219	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘𝑁)
32		eqid 2769	. . . . . . . . . . . . 13 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
33	2, 32, 30	ply1mulr 22353	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘(1o mPoly 𝑅))
34	23, 33	mgpplusg 20219	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
35	31, 34	eqtr3i 2794	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
36	35	a1i 11	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (+g‘𝑁) = (+g‘(mulGrp‘(1o mPoly 𝑅))))
37	36	oveqdr 7439	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑁)𝑦) = (𝑥(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑦))
38	6, 20, 22, 26, 28, 29, 37	mulgpropd 19181	. . . . . . 7 ⊢ (𝑅 ∈ Ring → ↑ = (.g‘(mulGrp‘(1o mPoly 𝑅))))
39	38	3ad2ant1 1149	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ↑ = (.g‘(mulGrp‘(1o mPoly 𝑅))))
40		eqidd 2770	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐷 = 𝐷)
41	3	vr1val 22320	. . . . . . 7 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)
42	41	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝑋 = ((1o mVar 𝑅)‘∅))
43	39, 40, 42	oveq123d 7432	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐷 ↑ 𝑋) = (𝐷(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘∅)))
44	43	oveq2d 7427	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) = (𝐶 · (𝐷(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘∅))))
45		psr1baslem 22313	. . . . . 6 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin}
46		coe1tm.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
47		eqid 2769	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
48		1on 8465	. . . . . . 7 ⊢ 1o ∈ On
49	48	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 1o ∈ On)
50		eqid 2769	. . . . . 6 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅)
51		simp1 1152	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝑅 ∈ Ring)
52		0lt1o 8488	. . . . . . 7 ⊢ ∅ ∈ 1o
53	52	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ∅ ∈ 1o)
54		simp3 1154	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0)
55	32, 45, 46, 47, 49, 23, 20, 50, 51, 53, 54	mplcoe3 22157	. . . . 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)‘∅)))
56	55	oveq2d 7427	. . . 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)‘∅))))
57	2, 32, 4	ply1vsca 22352	. . . . 5 ⊢ · = ( ·𝑠 ‘(1o mPoly 𝑅))
58		elsni 4611	. . . . . . . . . . 11 ⊢ (𝑏 ∈ {∅} → 𝑏 = ∅)
59		df1o2 8459	. . . . . . . . . . 11 ⊢ 1o = {∅}
60	58, 59	eleq2s 2887	. . . . . . . . . 10 ⊢ (𝑏 ∈ 1o → 𝑏 = ∅)
61	60	iftrued 4500	. . . . . . . . 9 ⊢ (𝑏 ∈ 1o → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
62	61	adantl 486	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑏 ∈ 1o) → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
63	62	mpteq2dva 5208	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (𝑏 ∈ 1o ↦ 𝐷))
64		fconstmpt 5724	. . . . . . 7 ⊢ (1o × {𝐷}) = (𝑏 ∈ 1o ↦ 𝐷)
65	63, 64	eqtr4di 2822	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (1o × {𝐷}))
66		fconst6g 6768	. . . . . . . 8 ⊢ (𝐷 ∈ ℕ0 → (1o × {𝐷}):1o⟶ℕ0)
67	14, 15	elmap 8868	. . . . . . . 8 ⊢ ((1o × {𝐷}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {𝐷}):1o⟶ℕ0)
68	66, 67	sylibr 237	. . . . . . 7 ⊢ (𝐷 ∈ ℕ0 → (1o × {𝐷}) ∈ (ℕ0 ↑m 1o))
69	68	3ad2ant3 1151	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (1o × {𝐷}) ∈ (ℕ0 ↑m 1o))
70	65, 69	eqeltrd 2869	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ∈ (ℕ0 ↑m 1o))
71		simp2 1153	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → 𝐶 ∈ 𝐾)
72	32, 57, 45, 47, 46, 1, 49, 51, 70, 71	mplmon2 22180	. . . 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 )))
73	44, 56, 72	3eqtr2d 2810	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) = (𝑦 ∈ (ℕ0 ↑m 1o) ↦ if(𝑦 = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
74		eqeq1 2773	. . . 4 ⊢ (𝑦 = (1o × {𝑥}) → (𝑦 = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0))))
75	74	ifbid 4516	. . 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 ))
76	18, 19, 73, 75	fmptco 7126	. 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 )))
77	65	adantr 485	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (1o × {𝐷}))
78	77	eqeq2d 2780	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1o × {𝑥}) = (1o × {𝐷})))
79		fveq1 6881	. . . . . . 7 ⊢ ((1o × {𝑥}) = (1o × {𝐷}) → ((1o × {𝑥})‘∅) = ((1o × {𝐷})‘∅))
80		vex 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
81	80	fvconst2 7203	. . . . . . . . 9 ⊢ (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
82	52, 81	mp1i 14	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥})‘∅) = 𝑥)
83		simpl3 1210	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐷 ∈ ℕ0)
84		fvconst2g 7201	. . . . . . . . 9 ⊢ ((𝐷 ∈ ℕ0 ∧ ∅ ∈ 1o) → ((1o × {𝐷})‘∅) = 𝐷)
85	83, 52, 84	sylancl 597	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝐷})‘∅) = 𝐷)
86	82, 85	eqeq12d 2785	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((1o × {𝑥})‘∅) = ((1o × {𝐷})‘∅) ↔ 𝑥 = 𝐷))
87	79, 86	imbitrid 247	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (1o × {𝐷}) → 𝑥 = 𝐷))
88		sneq 4604	. . . . . . 7 ⊢ (𝑥 = 𝐷 → {𝑥} = {𝐷})
89	88	xpeq2d 5692	. . . . . 6 ⊢ (𝑥 = 𝐷 → (1o × {𝑥}) = (1o × {𝐷}))
90	87, 89	impbid1 228	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (1o × {𝐷}) ↔ 𝑥 = 𝐷))
91	78, 90	bitrd 282	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ 𝑥 = 𝐷))
92	91	ifbid 4516	. . 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 ))
93	92	mpteq2dva 5208	. 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 )))
94	12, 76, 93	3eqtrd 2808	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ifcif 4492  {csn 4594   ↦ cmpt 5196   × cxp 5660   ∘ ccom 5666  Oncon0 6361  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  1_oc1o 8445   ↑_m cmap 8823  0cc0 11099  ℕ₀cn0 12503  Basecbs 17268  +_gcplusg 17309  ._rcmulr 17310   ·_𝑠 cvsca 17313  0_gc0g 17491  ._gcmg 19132  mulGrpcmgp 20215  1_rcur 20262  Ringcrg 20314   mVar cmvr 22023   mPoly cmpl 22024  var₁cv1 22304  Poly₁cpl1 22305  coe₁cco1 22306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-ofr 7676  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-sup 9401  df-oi 9471  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-fzo 13682  df-seq 14037  df-hash 14366  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-sca 17325  df-vsca 17326  df-ip 17327  df-tset 17328  df-ple 17329  df-ds 17331  df-hom 17333  df-cco 17334  df-0g 17493  df-gsum 17494  df-prds 17499  df-pws 17501  df-mre 17637  df-mrc 17638  df-acs 17640  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-mhm 18840  df-submnd 18841  df-grp 19002  df-minusg 19003  df-sbg 19004  df-mulg 19133  df-subg 19188  df-ghm 19283  df-cntz 19386  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-subrng 20630  df-subrg 20654  df-lmod 20960  df-lss 21030  df-psr 22027  df-mvr 22028  df-mpl 22029  df-opsr 22031  df-psr1 22308  df-vr1 22309  df-ply1 22310  df-coe1 22311

This theorem is referenced by:  coe1tmfv1  22403  coe1tmfv2  22404  coe1scl  22416  gsummoncoe1  22436  decpmatid  22895  monmatcollpw  22904  mp2pm2mplem4  22934  coe1mon  33821