Theorem coe1mul2 22390

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 6757	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (1o × {𝑘}):1o⟶ℕ0)
2		nn0ex 12501	. . . . . 6 ⊢ ℕ0 ∈ V
3		1on 8454	. . . . . . 7 ⊢ 1o ∈ On
4	3	elexi 3479	. . . . . 6 ⊢ 1o ∈ V
5	2, 4	elmap 8857	. . . . 5 ⊢ ((1o × {𝑘}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {𝑘}):1o⟶ℕ0)
6	1, 5	sylibr 237	. . . 4 ⊢ (𝑘 ∈ ℕ0 → (1o × {𝑘}) ∈ (ℕ0 ↑m 1o))
7	6	adantl 486	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (1o × {𝑘}) ∈ (ℕ0 ↑m 1o))
8		eqidd 2766	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})))
9		eqid 2765	. . . 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 22310	. . . 4 ⊢ 𝐵 = (Base‘(1o mPwSer 𝑅))
13		coe1mul2.u	. . . 4 ⊢ · = (.r‘𝑅)
14		coe1mul2.t	. . . . 5 ⊢ ∙ = (.r‘𝑆)
15	10, 9, 14	psr1mulr 22342	. . . 4 ⊢ ∙ = (.r‘(1o mPwSer 𝑅))
16		psr1baslem 22305	. . . 4 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin}
17		simp2 1153	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
18		simp3 1154	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
19	9, 12, 13, 15, 16, 17, 18	psrmulfval 22053	. . 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 5109	. . . . . 6 ⊢ (𝑏 = (1o × {𝑘}) → (𝑑 ∘r ≤ 𝑏 ↔ 𝑑 ∘r ≤ (1o × {𝑘})))
21	20	rabbidv 3424	. . . . 5 ⊢ (𝑏 = (1o × {𝑘}) → {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ 𝑏} = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})})
22		fvoveq1 7423	. . . . . 6 ⊢ (𝑏 = (1o × {𝑘}) → (𝐺‘(𝑏 ∘f − 𝑐)) = (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))
23	22	oveq2d 7416	. . . . 5 ⊢ (𝑏 = (1o × {𝑘}) → ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘f − 𝑐))) = ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐))))
24	21, 23	mpteq12dv 5192	. . . 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 7416	. . 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 7115	. 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 22366	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring)
28	11, 14	ringcl 20323	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
29	27, 28	syl3an1 1179	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
30		eqid 2765	. . . 4 ⊢ (coe1‘(𝐹 ∙ 𝐺)) = (coe1‘(𝐹 ∙ 𝐺))
31		eqid 2765	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))
32	30, 11, 10, 31	coe1fval3 22328	. . 3 ⊢ ((𝐹 ∙ 𝐺) ∈ 𝐵 → (coe1‘(𝐹 ∙ 𝐺)) = ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))))
33	29, 32	syl 18	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))))
34		eqid 2765	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
35		eqid 2765	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
36		simpl1 1208	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37		ringcmn 20356	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
38	36, 37	syl 18	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39		fzfi 13999	. . . . . 6 ⊢ (0...𝑘) ∈ Fin
40	39	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41		simpll1 1229	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42		simpll2 1230	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹 ∈ 𝐵)
43		eqid 2765	. . . . . . . . . 10 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
44	43, 11, 10, 34	coe1f2 22329	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹):ℕ0⟶(Base‘𝑅))
45	42, 44	syl 18	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1‘𝐹):ℕ0⟶(Base‘𝑅))
46		elfznn0 13639	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
47	46	adantl 486	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
48	45, 47	ffvelcdmd 7070	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅))
49		simpll3 1231	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺 ∈ 𝐵)
50		eqid 2765	. . . . . . . . . 10 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
51	50, 11, 10, 34	coe1f2 22329	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅))
52	49, 51	syl 18	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1‘𝐺):ℕ0⟶(Base‘𝑅))
53		fznn0sub 13575	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑘) → (𝑘 − 𝑥) ∈ ℕ0)
54	53	adantl 486	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘 − 𝑥) ∈ ℕ0)
55	52, 54	ffvelcdmd 7070	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅))
56	34, 13	ringcl 20323	. . . . . . 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 1394	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅))
58	57	fmpttd 7100	. . . . 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 3479	. . . . . . . . 9 ⊢ (0...𝑘) ∈ V
60	59	mptex 7211	. . . . . . . 8 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V
61		funmpt 6563	. . . . . . . 8 ⊢ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
62		fvex 6884	. . . . . . . 8 ⊢ (0g‘𝑅) ∈ V
63	60, 61, 62	3pm3.2i 1356	. . . . . . 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 8161	. . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
66	65	dmmptss 6232	. . . . . . . . 9 ⊢ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ⊆ (0...𝑘)
67	64, 66	sstri 3948	. . . . . . . 8 ⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘)
68	39, 67	pm3.2i 475	. . . . . . 7 ⊢ ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘))
69		suppssfifsupp 9328	. . . . . . 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 704	. . . . . 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 2765	. . . . . . 7 ⊢ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}
73	72	coe1mul2lem2 22389	. . . . . 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 486	. . . . 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 19977	. . . 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 5108	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑑 ∘r ≤ (1o × {𝑘}) ↔ 𝑐 ∘r ≤ (1o × {𝑘})))
77	76	elrab 3653	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↔ (𝑐 ∈ (ℕ0 ↑m 1o) ∧ 𝑐 ∘r ≤ (1o × {𝑘})))
78	77	simprbi 502	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} → 𝑐 ∘r ≤ (1o × {𝑘}))
79	78	adantl 486	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑐 ∘r ≤ (1o × {𝑘}))
80		simplr 780	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑘 ∈ ℕ0)
81		elrabi 3649	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} → 𝑐 ∈ (ℕ0 ↑m 1o))
82	81	adantl 486	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑐 ∈ (ℕ0 ↑m 1o))
83		coe1mul2lem1 22388	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑐 ∈ (ℕ0 ↑m 1o)) → (𝑐 ∘r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
84	80, 82, 83	syl2anc 595	. . . . . . . 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 235	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
86		eqidd 2766	. . . . . . 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 2766	. . . . . . 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 6871	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐹)‘𝑥) = ((coe1‘𝐹)‘(𝑐‘∅)))
89		oveq2 7408	. . . . . . . . 9 ⊢ (𝑥 = (𝑐‘∅) → (𝑘 − 𝑥) = (𝑘 − (𝑐‘∅)))
90	89	fveq2d 6875	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) = ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))
91	88, 90	oveq12d 7418	. . . . . . 7 ⊢ (𝑥 = (𝑐‘∅) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) = (((coe1‘𝐹)‘(𝑐‘∅)) · ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))))
92	85, 86, 87, 91	fmptco 7115	. . . . . 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 1230	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝐹 ∈ 𝐵)
94	43	fvcoe1 22327	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑐 ∈ (ℕ0 ↑m 1o)) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅)))
95	93, 82, 94	syl2anc 595	. . . . . . . 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 8448	. . . . . . . . . . . . . 14 ⊢ 1o = {∅}
97		0ex 5262	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
98	96, 2, 97	mapsnconst 8878	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (ℕ0 ↑m 1o) → 𝑐 = (1o × {(𝑐‘∅)}))
99	82, 98	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝑐 = (1o × {(𝑐‘∅)}))
100	99	oveq2d 7416	. . . . . . . . . . 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 3461	. . . . . . . . . . . . 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 6886	. . . . . . . . . . . 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 7694	. . . . . . . . . . 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 2800	. . . . . . . . . 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 6875	. . . . . . . . 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 1231	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝐺 ∈ 𝐵)
109		fznn0sub 13575	. . . . . . . . . . 11 ⊢ ((𝑐‘∅) ∈ (0...𝑘) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
110	85, 109	syl 18	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
111	50	coe1fv 22326	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
112	108, 110, 111	syl2anc 595	. . . . . . . . 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 2803	. . . . . . . 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 7418	. . . . . . 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 5198	. . . . . 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 2803	. . . . 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 7416	. . . 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 2800	. . 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 5198	. 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 2810	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  {csn 4585   class class class wbr 5105   ↦ cmpt 5186   × cxp 5650  dom cdm 5652   ∘ ccom 5656  Oncon0 6350  Fun wfun 6519  ⟶wf 6521  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   ∘_f cof 7662   ∘_r cofr 7663   supp csupp 8144  1_oc1o 8434   ↑_m cmap 8812  Fincfn 8931   finSupp cfsupp 9309  0cc0 11088   ≤ cle 11232   − cmin 11429  ℕ₀cn0 12495  ...cfz 13526  Basecbs 17259  ._rcmulr 17301  0_gc0g 17482   Σ_g cgsu 17483  CMndccmn 19841  Ringcrg 20306   mPwSer cmps 22014  PwSer₁cps1 22295  coe₁cco1 22298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-ofr 7665  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-hom 17324  df-cco 17325  df-0g 17484  df-gsum 17485  df-prds 17490  df-pws 17492  df-mre 17628  df-mrc 17629  df-acs 17631  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-submnd 18832  df-grp 18993  df-minusg 18994  df-mulg 19125  df-ghm 19275  df-cntz 19378  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-psr 22019  df-opsr 22023  df-psr1 22300  df-coe1 22303

This theorem is referenced by:  coe1mul  22391