Theorem coe1mul2 22262

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 6723	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (1o × {𝑘}):1o⟶ℕ0)
2		nn0ex 12441	. . . . . 6 ⊢ ℕ0 ∈ V
3		1on 8414	. . . . . . 7 ⊢ 1o ∈ On
4	3	elexi 3455	. . . . . 6 ⊢ 1o ∈ V
5	2, 4	elmap 8816	. . . . 5 ⊢ ((1o × {𝑘}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {𝑘}):1o⟶ℕ0)
6	1, 5	sylibr 235	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (1o × {𝑘}) ∈ (ℕ0 ↑m 1o))
7	6	adantl 482	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (1o × {𝑘}) ∈ (ℕ0 ↑m 1o))
8		eqidd 2741	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})))
9		eqid 2740	. . . 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 22182	. . . 4 ⊢ 𝐵 = (Base‘(1o mPwSer 𝑅))
13		coe1mul2.u	. . . 4 ⊢ · = (.r‘𝑅)
14		coe1mul2.t	. . . . 5 ⊢ ∙ = (.r‘𝑆)
15	10, 9, 14	psr1mulr 22214	. . . 4 ⊢ ∙ = (.r‘(1o mPwSer 𝑅))
16		psr1baslem 22177	. . . 4 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin}
17		simp2 1143	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
18		simp3 1144	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
19	9, 12, 13, 15, 16, 17, 18	psrmulfval 21925	. . 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 5083	. . . . . 6 ⊢ (𝑏 = (1o × {𝑘}) → (𝑑 ∘r ≤ 𝑏 ↔ 𝑑 ∘r ≤ (1o × {𝑘})))
21	20	rabbidv 3399	. . . . 5 ⊢ (𝑏 = (1o × {𝑘}) → {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ 𝑏} = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})})
22		fvoveq1 7386	. . . . . 6 ⊢ (𝑏 = (1o × {𝑘}) → (𝐺‘(𝑏 ∘f − 𝑐)) = (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))
23	22	oveq2d 7379	. . . . 5 ⊢ (𝑏 = (1o × {𝑘}) → ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘f − 𝑐))) = ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐))))
24	21, 23	mpteq12dv 5166	. . . 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 7379	. . 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 7078	. 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 22238	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring)
28	11, 14	ringcl 20229	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
29	27, 28	syl3an1 1169	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
30		eqid 2740	. . . 4 ⊢ (coe1‘(𝐹 ∙ 𝐺)) = (coe1‘(𝐹 ∙ 𝐺))
31		eqid 2740	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))
32	30, 11, 10, 31	coe1fval3 22200	. . 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 2740	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
35		eqid 2740	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
36		simpl1 1198	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37		ringcmn 20261	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
38	36, 37	syl 17	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39		fzfi 13932	. . . . . 6 ⊢ (0...𝑘) ∈ Fin
40	39	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41		simpll1 1219	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42		simpll2 1220	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹 ∈ 𝐵)
43		eqid 2740	. . . . . . . . . 10 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
44	43, 11, 10, 34	coe1f2 22201	. . . . . . . . 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 13572	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
47	46	adantl 482	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
48	45, 47	ffvelcdmd 7033	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅))
49		simpll3 1221	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺 ∈ 𝐵)
50		eqid 2740	. . . . . . . . . 10 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
51	50, 11, 10, 34	coe1f2 22201	. . . . . . . . 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 13508	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑘) → (𝑘 − 𝑥) ∈ ℕ0)
54	53	adantl 482	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘 − 𝑥) ∈ ℕ0)
55	52, 54	ffvelcdmd 7033	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅))
56	34, 13	ringcl 20229	. . . . . . 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 1379	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅))
58	57	fmpttd 7063	. . . . 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 3455	. . . . . . . . 9 ⊢ (0...𝑘) ∈ V
60	59	mptex 7174	. . . . . . . 8 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V
61		funmpt 6530	. . . . . . . 8 ⊢ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
62		fvex 6847	. . . . . . . 8 ⊢ (0g‘𝑅) ∈ V
63	60, 61, 62	3pm3.2i 1346	. . . . . . 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 8124	. . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
66	65	dmmptss 6199	. . . . . . . . 9 ⊢ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ⊆ (0...𝑘)
67	64, 66	sstri 3931	. . . . . . . 8 ⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘)
68	39, 67	pm3.2i 471	. . . . . . 7 ⊢ ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘))
69		suppssfifsupp 9290	. . . . . . 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 698	. . . . . 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 2740	. . . . . . 7 ⊢ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}
73	72	coe1mul2lem2 22261	. . . . . 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 482	. . . . 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 19889	. . . 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 5082	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑑 ∘r ≤ (1o × {𝑘}) ↔ 𝑐 ∘r ≤ (1o × {𝑘})))
77	76	elrab 3636	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↔ (𝑐 ∈ (ℕ0 ↑m 1o) ∧ 𝑐 ∘r ≤ (1o × {𝑘})))
78	77	simprbi 498	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} → 𝑐 ∘r ≤ (1o × {𝑘}))
79	78	adantl 482	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑐 ∘r ≤ (1o × {𝑘}))
80		simplr 774	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑘 ∈ ℕ0)
81		elrabi 3632	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} → 𝑐 ∈ (ℕ0 ↑m 1o))
82	81	adantl 482	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑐 ∈ (ℕ0 ↑m 1o))
83		coe1mul2lem1 22260	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑐 ∈ (ℕ0 ↑m 1o)) → (𝑐 ∘r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
84	80, 82, 83	syl2anc 590	. . . . . . . 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 233	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
86		eqidd 2741	. . . . . . 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 2741	. . . . . . 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 6834	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐹)‘𝑥) = ((coe1‘𝐹)‘(𝑐‘∅)))
89		oveq2 7371	. . . . . . . . 9 ⊢ (𝑥 = (𝑐‘∅) → (𝑘 − 𝑥) = (𝑘 − (𝑐‘∅)))
90	89	fveq2d 6838	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) = ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))
91	88, 90	oveq12d 7381	. . . . . . 7 ⊢ (𝑥 = (𝑐‘∅) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) = (((coe1‘𝐹)‘(𝑐‘∅)) · ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))))
92	85, 86, 87, 91	fmptco 7078	. . . . . 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 1220	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝐹 ∈ 𝐵)
94	43	fvcoe1 22199	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑐 ∈ (ℕ0 ↑m 1o)) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅)))
95	93, 82, 94	syl2anc 590	. . . . . . . 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 8409	. . . . . . . . . . . . . 14 ⊢ 1o = {∅}
97		0ex 5236	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
98	96, 2, 97	mapsnconst 8837	. . . . . . . . . . . . 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 7379	. . . . . . . . . . 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 3436	. . . . . . . . . . . . 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 6849	. . . . . . . . . . . 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 7657	. . . . . . . . . . 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 2775	. . . . . . . . . 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 6838	. . . . . . . . 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 1221	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝐺 ∈ 𝐵)
109		fznn0sub 13508	. . . . . . . . . . 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 22198	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
112	108, 110, 111	syl2anc 590	. . . . . . . . 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 2778	. . . . . . . 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 7381	. . . . . . 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 5172	. . . . . 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 2778	. . . . 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 7379	. . . 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 2775	. . 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 5172	. 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 2785	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432   ⊆ wss 3890  ∅c0 4268  {csn 4562   class class class wbr 5079   ↦ cmpt 5160   × cxp 5623  dom cdm 5625   ∘ ccom 5629  Oncon0 6317  Fun wfun 6486  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363   ∘_f cof 7625   ∘_r cofr 7626   supp csupp 8107  1_oc1o 8395   ↑_m cmap 8770  Fincfn 8890   finSupp cfsupp 9271  0cc0 11036   ≤ cle 11178   − cmin 11375  ℕ₀cn0 12435  ...cfz 13459  Basecbs 17177  ._rcmulr 17219  0_gc0g 17400   Σ_g cgsu 17401  CMndccmn 19753  Ringcrg 20212   mPwSer cmps 21886  PwSer₁cps1 22167  coe₁cco1 22170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-ofr 7628  df-om 7814  df-1st 7938  df-2nd 7939  df-supp 8108  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-map 8772  df-pm 8773  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9272  df-sup 9352  df-oi 9422  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-uz 12787  df-fz 13460  df-fzo 13607  df-seq 13962  df-hash 14291  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-ip 17236  df-tset 17237  df-ple 17238  df-ds 17240  df-hom 17242  df-cco 17243  df-0g 17402  df-gsum 17403  df-prds 17408  df-pws 17410  df-mre 17546  df-mrc 17547  df-acs 17549  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-mhm 18749  df-submnd 18750  df-grp 18910  df-minusg 18911  df-mulg 19042  df-ghm 19186  df-cntz 19290  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-ring 20214  df-psr 21891  df-opsr 21895  df-psr1 22172  df-coe1 22175

This theorem is referenced by:  coe1mul  22263