Theorem coe1tmmul2 20446

Description: Coefficient vector of a polynomial multiplied on the right by a 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)
coe1tmmul.b	⊢ 𝐵 = (Base‘𝑃)
coe1tmmul.t	⊢ ∙ = (.r‘𝑃)
coe1tmmul.u	⊢ × = (.r‘𝑅)
coe1tmmul.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
coe1tmmul.r	⊢ (𝜑 → 𝑅 ∈ Ring)
coe1tmmul.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)
coe1tmmul.d	⊢ (𝜑 → 𝐷 ∈ ℕ0)

Assertion

Ref	Expression
coe1tmmul2	⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))

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

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem coe1tmmul2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1tmmul.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		coe1tmmul.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		coe1tmmul.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
4		coe1tmmul.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
5		coe1tm.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
6		coe1tm.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
7		coe1tm.x	. . . . 5 ⊢ 𝑋 = (var1‘𝑅)
8		coe1tm.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑃)
9		coe1tm.n	. . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃)
10		coe1tm.e	. . . . 5 ⊢ ↑ = (.g‘𝑁)
11		coe1tmmul.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
12	5, 6, 7, 8, 9, 10, 11	ply1tmcl 20442	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
13	1, 3, 4, 12	syl3anc 1367	. . 3 ⊢ (𝜑 → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
14		coe1tmmul.t	. . . 4 ⊢ ∙ = (.r‘𝑃)
15		coe1tmmul.u	. . . 4 ⊢ × = (.r‘𝑅)
16	6, 14, 15, 11	coe1mul 20440	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵) → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))))
17	1, 2, 13, 16	syl3anc 1367	. 2 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))))
18		eqeq2 2835	. . . 4 ⊢ ((((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
19		eqeq2 2835	. . . 4 ⊢ ( 0 = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
20		coe1tm.z	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
21	1	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑅 ∈ Ring)
22		ringmnd 19308	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
23	21, 22	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑅 ∈ Mnd)
24		ovex 7191	. . . . . . . 8 ⊢ (0...𝑥) ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (0...𝑥) ∈ V)
26		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ≤ 𝑥)
27	4	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℕ0)
28		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℕ0)
29		nn0sub 11950	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → (𝐷 ≤ 𝑥 ↔ (𝑥 − 𝐷) ∈ ℕ0))
30	27, 28, 29	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝐷 ≤ 𝑥 ↔ (𝑥 − 𝐷) ∈ ℕ0))
31	26, 30	mpbid 234	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ∈ ℕ0)
32	27	nn0ge0d 11961	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐷)
33		nn0re 11909	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ)
34	33	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ)
35	4	nn0red 11959	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ)
36	35	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℝ)
37	34, 36	subge02d 11234	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (0 ≤ 𝐷 ↔ (𝑥 − 𝐷) ≤ 𝑥))
38	32, 37	mpbid 234	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ≤ 𝑥)
39		fznn0 13002	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → ((𝑥 − 𝐷) ∈ (0...𝑥) ↔ ((𝑥 − 𝐷) ∈ ℕ0 ∧ (𝑥 − 𝐷) ≤ 𝑥)))
40	39	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑥 − 𝐷) ∈ (0...𝑥) ↔ ((𝑥 − 𝐷) ∈ ℕ0 ∧ (𝑥 − 𝐷) ≤ 𝑥)))
41	31, 38, 40	mpbir2and 711	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ∈ (0...𝑥))
42	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43		eqid 2823	. . . . . . . . . . . . 13 ⊢ (coe1‘𝐴) = (coe1‘𝐴)
44	43, 11, 6, 5	coe1f 20381	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (coe1‘𝐴):ℕ0⟶𝐾)
45	2, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘𝐴):ℕ0⟶𝐾)
46	45	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘𝐴):ℕ0⟶𝐾)
47		elfznn0 13003	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
48	47	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
49	46, 48	ffvelrnd 6854	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘𝐴)‘𝑦) ∈ 𝐾)
50		eqid 2823	. . . . . . . . . . . . 13 ⊢ (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐷 ↑ 𝑋)))
51	50, 11, 6, 5	coe1f 20381	. . . . . . . . . . . 12 ⊢ ((𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
52	13, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
53	52	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
54		fznn0sub 12942	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℕ0)
55	54	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − 𝑦) ∈ ℕ0)
56	53, 55	ffvelrnd 6854	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) ∈ 𝐾)
57	5, 15	ringcl 19313	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) ∈ 𝐾) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) ∈ 𝐾)
58	42, 49, 56, 57	syl3anc 1367	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) ∈ 𝐾)
59	58	fmpttd 6881	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))):(0...𝑥)⟶𝐾)
60	1	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝑅 ∈ Ring)
61	3	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐶 ∈ 𝐾)
62	4	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐷 ∈ ℕ0)
63		eldifi 4105	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) → 𝑦 ∈ (0...𝑥))
64	63, 54	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) → (𝑥 − 𝑦) ∈ ℕ0)
65	64	adantl 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (𝑥 − 𝑦) ∈ ℕ0)
66		eldifsn 4721	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥 − 𝐷)))
67		simplrl 775	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
68	67	nn0cnd 11960	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
69	47	nn0cnd 11960	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
70	69	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
71	68, 70	nncand 11004	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥 − 𝑦)) = 𝑦)
72	71	eqcomd 2829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥 − 𝑦)))
73		oveq2 7166	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 = (𝑥 − 𝑦) → (𝑥 − 𝐷) = (𝑥 − (𝑥 − 𝑦)))
74	73	eqeq2d 2834	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = (𝑥 − 𝑦) → (𝑦 = (𝑥 − 𝐷) ↔ 𝑦 = (𝑥 − (𝑥 − 𝑦))))
75	72, 74	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥 − 𝑦) → 𝑦 = (𝑥 − 𝐷)))
76	75	necon3d 3039	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥 − 𝐷) → 𝐷 ≠ (𝑥 − 𝑦)))
77	76	impr 457	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥 − 𝐷))) → 𝐷 ≠ (𝑥 − 𝑦))
78	66, 77	sylan2b 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐷 ≠ (𝑥 − 𝑦))
79	20, 5, 6, 7, 8, 9, 10, 60, 61, 62, 65, 78	coe1tmfv2 20445	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = 0 )
80	79	oveq2d 7174	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘𝑦) × 0 ))
81	5, 15, 20	ringrz 19340	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘𝑦) ∈ 𝐾) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
82	42, 49, 81	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
83	63, 82	sylan2 594	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
84	80, 83	eqtrd 2858	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
85	84, 25	suppss2 7866	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) supp 0 ) ⊆ {(𝑥 − 𝐷)})
86	5, 20, 23, 25, 41, 59, 85	gsumpt 19084	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)))
87		fveq2 6672	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 𝐷) → ((coe1‘𝐴)‘𝑦) = ((coe1‘𝐴)‘(𝑥 − 𝐷)))
88		oveq2 7166	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 − 𝐷) → (𝑥 − 𝑦) = (𝑥 − (𝑥 − 𝐷)))
89	88	fveq2d 6676	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 𝐷) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))))
90	87, 89	oveq12d 7176	. . . . . . . 8 ⊢ (𝑦 = (𝑥 − 𝐷) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
91		eqid 2823	. . . . . . . 8 ⊢ (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))
92		ovex 7191	. . . . . . . 8 ⊢ (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))) ∈ V
93	90, 91, 92	fvmpt 6770	. . . . . . 7 ⊢ ((𝑥 − 𝐷) ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
94	41, 93	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
95	28	nn0cnd 11960	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℂ)
96	27	nn0cnd 11960	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℂ)
97	95, 96	nncand 11004	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − (𝑥 − 𝐷)) = 𝐷)
98	97	fveq2d 6676	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷))
99	3	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐶 ∈ 𝐾)
100	20, 5, 6, 7, 8, 9, 10	coe1tmfv1 20444	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
101	21, 99, 27, 100	syl3anc 1367	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
102	98, 101	eqtrd 2858	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))) = 𝐶)
103	102	oveq2d 7174	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
104	86, 94, 103	3eqtrd 2862	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
105	104	anassrs 470	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
106	1	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
107	3	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐶 ∈ 𝐾)
108	4	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
109	54	ad2antll 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ∈ ℕ0)
110	54	nn0red 11959	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℝ)
111	110	ad2antll 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ∈ ℝ)
112	33	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
113	35	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
114	47	ad2antll 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115	114	nn0ge0d 11961	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
116	47	nn0red 11959	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117	116	ad2antll 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118	112, 117	subge02d 11234	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥 − 𝑦) ≤ 𝑥))
119	115, 118	mpbid 234	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ≤ 𝑥)
120		simprl 769	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ¬ 𝐷 ≤ 𝑥)
121	112, 113	ltnled 10789	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷 ≤ 𝑥))
122	120, 121	mpbird 259	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123	111, 112, 113, 119, 122	lelttrd 10800	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) < 𝐷)
124	111, 123	gtned 10777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥 − 𝑦))
125	20, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124	coe1tmfv2 20445	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = 0 )
126	125	oveq2d 7174	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘𝑦) × 0 ))
127	45	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (coe1‘𝐴):ℕ0⟶𝐾)
128	127, 114	ffvelrnd 6854	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ((coe1‘𝐴)‘𝑦) ∈ 𝐾)
129	106, 128, 81	syl2anc 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
130	126, 129	eqtrd 2858	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
131	130	anassrs 470	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
132	131	mpteq2dva 5163	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133	132	oveq2d 7174	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
134	1, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mnd)
135	20	gsumz 18002	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136	134, 24, 135	sylancl 588	. . . . . 6 ⊢ (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137	136	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138	133, 137	eqtrd 2858	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = 0 )
139	18, 19, 105, 138	ifbothda 4506	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))
140	139	mpteq2dva 5163	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
141	17, 140	eqtrd 2858	1 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  Vcvv 3496   ∖ cdif 3935  ifcif 4469  {csn 4569   class class class wbr 5068   ↦ cmpt 5148  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  ℂcc 10537  ℝcr 10538  0cc0 10539   < clt 10677   ≤ cle 10678   − cmin 10872  ℕ₀cn0 11900  ...cfz 12895  Basecbs 16485  ._rcmulr 16568   ·_𝑠 cvsca 16571  0_gc0g 16715   Σ_g cgsu 16716  Mndcmnd 17913  ._gcmg 18226  mulGrpcmgp 19241  Ringcrg 19299  var₁cv1 20346  Poly₁cpl1 20347  coe₁cco1 20348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-ofr 7412  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-tset 16586  df-ple 16587  df-0g 16717  df-gsum 16718  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-ghm 18358  df-cntz 18449  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-ring 19301  df-subrg 19535  df-lmod 19638  df-lss 19706  df-psr 20138  df-mvr 20139  df-mpl 20140  df-opsr 20142  df-psr1 20350  df-vr1 20351  df-ply1 20352  df-coe1 20353

This theorem is referenced by:  coe1tmmul2fv  20448  coe1sclmul2  20454