Theorem coe1mul2 22223

Description: The coefficient vector of multiplication in the univariate power series ring. (Contributed by Stefan O'Rear, 25-Mar-2015.)

Hypotheses

Ref	Expression
coe1mul2.s	⊢ 𝑆 = (PwSer1‘𝑅)
coe1mul2.t	⊢ ∙ = (.r‘𝑆)
coe1mul2.u	⊢ · = (.r‘𝑅)
coe1mul2.b	⊢ 𝐵 = (Base‘𝑆)

Assertion

Ref	Expression
coe1mul2	⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))))))

Distinct variable groups:   𝑥,𝑘,𝐵   𝑘,𝐹,𝑥   · ,𝑘,𝑥   𝑘,𝐺,𝑥   𝑅,𝑘,𝑥   ∙ ,𝑘

Allowed substitution hints:   𝑆(𝑥,𝑘)   ∙ (𝑥)

Proof of Theorem coe1mul2

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

Step	Hyp	Ref	Expression
1		fconst6g 6731	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (1o × {𝑘}):1o⟶ℕ0)
2		nn0ex 12419	. . . . . 6 ⊢ ℕ0 ∈ V
3		1on 8419	. . . . . . 7 ⊢ 1o ∈ On
4	3	elexi 3465	. . . . . 6 ⊢ 1o ∈ V
5	2, 4	elmap 8821	. . . . 5 ⊢ ((1o × {𝑘}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {𝑘}):1o⟶ℕ0)
6	1, 5	sylibr 234	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (1o × {𝑘}) ∈ (ℕ0 ↑m 1o))
7	6	adantl 481	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (1o × {𝑘}) ∈ (ℕ0 ↑m 1o))
8		eqidd 2738	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})))
9		eqid 2737	. . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅)
10		coe1mul2.s	. . . . 5 ⊢ 𝑆 = (PwSer1‘𝑅)
11		coe1mul2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
12	10, 11, 9	psr1bas2 22142	. . . 4 ⊢ 𝐵 = (Base‘(1o mPwSer 𝑅))
13		coe1mul2.u	. . . 4 ⊢ · = (.r‘𝑅)
14		coe1mul2.t	. . . . 5 ⊢ ∙ = (.r‘𝑆)
15	10, 9, 14	psr1mulr 22175	. . . 4 ⊢ ∙ = (.r‘(1o mPwSer 𝑅))
16		psr1baslem 22137	. . . 4 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin}
17		simp2 1138	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
18		simp3 1139	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
19	9, 12, 13, 15, 16, 17, 18	psrmulfval 21911	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝑏 ∈ (ℕ0 ↑m 1o) ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ 𝑏} ↦ ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘f − 𝑐)))))))
20		breq2 5104	. . . . . 6 ⊢ (𝑏 = (1o × {𝑘}) → (𝑑 ∘r ≤ 𝑏 ↔ 𝑑 ∘r ≤ (1o × {𝑘})))
21	20	rabbidv 3408	. . . . 5 ⊢ (𝑏 = (1o × {𝑘}) → {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ 𝑏} = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})})
22		fvoveq1 7391	. . . . . 6 ⊢ (𝑏 = (1o × {𝑘}) → (𝐺‘(𝑏 ∘f − 𝑐)) = (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))
23	22	oveq2d 7384	. . . . 5 ⊢ (𝑏 = (1o × {𝑘}) → ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘f − 𝑐))) = ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐))))
24	21, 23	mpteq12dv 5187	. . . 4 ⊢ (𝑏 = (1o × {𝑘}) → (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ 𝑏} ↦ ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘f − 𝑐)))) = (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))))
25	24	oveq2d 7384	. . 3 ⊢ (𝑏 = (1o × {𝑘}) → (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ 𝑏} ↦ ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘f − 𝑐))))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐))))))
26	7, 8, 19, 25	fmptco 7084	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))))))
27	10	psr1ring 22199	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring)
28	11, 14	ringcl 20197	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
29	27, 28	syl3an1 1164	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
30		eqid 2737	. . . 4 ⊢ (coe1‘(𝐹 ∙ 𝐺)) = (coe1‘(𝐹 ∙ 𝐺))
31		eqid 2737	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))
32	30, 11, 10, 31	coe1fval3 22161	. . 3 ⊢ ((𝐹 ∙ 𝐺) ∈ 𝐵 → (coe1‘(𝐹 ∙ 𝐺)) = ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))))
33	29, 32	syl 17	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))))
34		eqid 2737	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
35		eqid 2737	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
36		simpl1 1193	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37		ringcmn 20229	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
38	36, 37	syl 17	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39		fzfi 13907	. . . . . 6 ⊢ (0...𝑘) ∈ Fin
40	39	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41		simpll1 1214	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42		simpll2 1215	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹 ∈ 𝐵)
43		eqid 2737	. . . . . . . . . 10 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
44	43, 11, 10, 34	coe1f2 22162	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹):ℕ0⟶(Base‘𝑅))
45	42, 44	syl 17	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1‘𝐹):ℕ0⟶(Base‘𝑅))
46		elfznn0 13548	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
47	46	adantl 481	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
48	45, 47	ffvelcdmd 7039	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅))
49		simpll3 1216	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺 ∈ 𝐵)
50		eqid 2737	. . . . . . . . . 10 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
51	50, 11, 10, 34	coe1f2 22162	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅))
52	49, 51	syl 17	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1‘𝐺):ℕ0⟶(Base‘𝑅))
53		fznn0sub 13484	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑘) → (𝑘 − 𝑥) ∈ ℕ0)
54	53	adantl 481	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘 − 𝑥) ∈ ℕ0)
55	52, 54	ffvelcdmd 7039	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅))
56	34, 13	ringcl 20197	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅)) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅))
57	41, 48, 55, 56	syl3anc 1374	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅))
58	57	fmpttd 7069	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))):(0...𝑘)⟶(Base‘𝑅))
59	39	elexi 3465	. . . . . . . . 9 ⊢ (0...𝑘) ∈ V
60	59	mptex 7179	. . . . . . . 8 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V
61		funmpt 6538	. . . . . . . 8 ⊢ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
62		fvex 6855	. . . . . . . 8 ⊢ (0g‘𝑅) ∈ V
63	60, 61, 62	3pm3.2i 1341	. . . . . . 7 ⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∧ (0g‘𝑅) ∈ V)
64		suppssdm 8129	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
65		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
66	65	dmmptss 6207	. . . . . . . . 9 ⊢ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ⊆ (0...𝑘)
67	64, 66	sstri 3945	. . . . . . . 8 ⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘)
68	39, 67	pm3.2i 470	. . . . . . 7 ⊢ ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘))
69		suppssfifsupp 9295	. . . . . . 7 ⊢ ((((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V ∧ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∧ (0g‘𝑅) ∈ V) ∧ ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘))) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) finSupp (0g‘𝑅))
70	63, 68, 69	mp2an 693	. . . . . 6 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) finSupp (0g‘𝑅)
71	70	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) finSupp (0g‘𝑅))
72		eqid 2737	. . . . . . 7 ⊢ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}
73	72	coe1mul2lem2 22222	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}–1-1-onto→(0...𝑘))
74	73	adantl 481	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)):{𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}–1-1-onto→(0...𝑘))
75	34, 35, 38, 40, 58, 71, 74	gsumf1o 19857	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))) = (𝑅 Σg ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))))
76		breq1 5103	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑑 ∘r ≤ (1o × {𝑘}) ↔ 𝑐 ∘r ≤ (1o × {𝑘})))
77	76	elrab 3648	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↔ (𝑐 ∈ (ℕ0 ↑m 1o) ∧ 𝑐 ∘r ≤ (1o × {𝑘})))
78	77	simprbi 497	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} → 𝑐 ∘r ≤ (1o × {𝑘}))
79	78	adantl 481	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑐 ∘r ≤ (1o × {𝑘}))
80		simplr 769	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑘 ∈ ℕ0)
81		elrabi 3644	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} → 𝑐 ∈ (ℕ0 ↑m 1o))
82	81	adantl 481	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑐 ∈ (ℕ0 ↑m 1o))
83		coe1mul2lem1 22221	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑐 ∈ (ℕ0 ↑m 1o)) → (𝑐 ∘r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
84	80, 82, 83	syl2anc 585	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝑐 ∘r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
85	79, 84	mpbid 232	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
86		eqidd 2738	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)) = (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))
87		eqidd 2738	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))))
88		fveq2 6842	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐹)‘𝑥) = ((coe1‘𝐹)‘(𝑐‘∅)))
89		oveq2 7376	. . . . . . . . 9 ⊢ (𝑥 = (𝑐‘∅) → (𝑘 − 𝑥) = (𝑘 − (𝑐‘∅)))
90	89	fveq2d 6846	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) = ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))
91	88, 90	oveq12d 7386	. . . . . . 7 ⊢ (𝑥 = (𝑐‘∅) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) = (((coe1‘𝐹)‘(𝑐‘∅)) · ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))))
92	85, 86, 87, 91	fmptco 7084	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (𝑐‘∅))) = (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (((coe1‘𝐹)‘(𝑐‘∅)) · ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))))
93		simpll2 1215	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝐹 ∈ 𝐵)
94	43	fvcoe1 22160	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑐 ∈ (ℕ0 ↑m 1o)) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅)))
95	93, 82, 94	syl2anc 585	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅)))
96		df1o2 8414	. . . . . . . . . . . . . 14 ⊢ 1o = {∅}
97		0ex 5254	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
98	96, 2, 97	mapsnconst 8842	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (ℕ0 ↑m 1o) → 𝑐 = (1o × {(𝑐‘∅)}))
99	82, 98	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑐 = (1o × {(𝑐‘∅)}))
100	99	oveq2d 7384	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f − 𝑐) = ((1o × {𝑘}) ∘f − (1o × {(𝑐‘∅)})))
101	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 1o ∈ On)
102		vex 3446	. . . . . . . . . . . . 13 ⊢ 𝑘 ∈ V
103	102	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑘 ∈ V)
104		fvexd 6857	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ V)
105	101, 103, 104	ofc12 7662	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f − (1o × {(𝑐‘∅)})) = (1o × {(𝑘 − (𝑐‘∅))}))
106	100, 105	eqtrd 2772	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → ((1o × {𝑘}) ∘f − 𝑐) = (1o × {(𝑘 − (𝑐‘∅))}))
107	106	fveq2d 6846	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f − 𝑐)) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
108		simpll3 1216	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝐺 ∈ 𝐵)
109		fznn0sub 13484	. . . . . . . . . . 11 ⊢ ((𝑐‘∅) ∈ (0...𝑘) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
110	85, 109	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
111	50	coe1fv 22159	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
112	108, 110, 111	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
113	107, 112	eqtr4d 2775	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝐺‘((1o × {𝑘}) ∘f − 𝑐)) = ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))
114	95, 113	oveq12d 7386	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐))) = (((coe1‘𝐹)‘(𝑐‘∅)) · ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))))
115	114	mpteq2dva 5193	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))) = (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (((coe1‘𝐹)‘(𝑐‘∅)) · ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))))
116	92, 115	eqtr4d 2775	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (𝑐‘∅))) = (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))))
117	116	oveq2d 7384	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∘ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ (𝑐‘∅)))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐))))))
118	75, 117	eqtrd 2772	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐))))))
119	118	mpteq2dva 5193	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))))) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↦ ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))))))
120	26, 33, 119	3eqtr4d 2782	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  {csn 4582   class class class wbr 5100   ↦ cmpt 5181   × cxp 5630  dom cdm 5632   ∘ ccom 5636  Oncon0 6325  Fun wfun 6494  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   ∘_f cof 7630   ∘_r cofr 7631   supp csupp 8112  1_oc1o 8400   ↑_m cmap 8775  Fincfn 8895   finSupp cfsupp 9276  0cc0 11038   ≤ cle 11179   − cmin 11376  ℕ₀cn0 12413  ...cfz 13435  Basecbs 17148  ._rcmulr 17190  0_gc0g 17371   Σ_g cgsu 17372  CMndccmn 19721  Ringcrg 20180   mPwSer cmps 21872  PwSer₁cps1 22127  coe₁cco1 22130

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-ofr 7633  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-sup 9357  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-fzo 13583  df-seq 13937  df-hash 14266  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-hom 17213  df-cco 17214  df-0g 17373  df-gsum 17374  df-prds 17379  df-pws 17381  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-submnd 18721  df-grp 18878  df-minusg 18879  df-mulg 19010  df-ghm 19154  df-cntz 19258  df-cmn 19723  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-psr 21877  df-opsr 21881  df-psr1 22132  df-coe1 22135

This theorem is referenced by:  coe1mul  22224