Theorem coe1mul2 22287

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 6797	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (1o × {𝑘}):1o⟶ℕ0)
2		nn0ex 12529	. . . . . 6 ⊢ ℕ0 ∈ V
3		1on 8516	. . . . . . 7 ⊢ 1o ∈ On
4	3	elexi 3500	. . . . . 6 ⊢ 1o ∈ V
5	2, 4	elmap 8909	. . . . 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 2735	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})))
9		eqid 2734	. . . 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 22206	. . . 4 ⊢ 𝐵 = (Base‘(1o mPwSer 𝑅))
13		coe1mul2.u	. . . 4 ⊢ · = (.r‘𝑅)
14		coe1mul2.t	. . . . 5 ⊢ ∙ = (.r‘𝑆)
15	10, 9, 14	psr1mulr 22239	. . . 4 ⊢ ∙ = (.r‘(1o mPwSer 𝑅))
16		psr1baslem 22201	. . . 4 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin}
17		simp2 1136	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
18		simp3 1137	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
19	9, 12, 13, 15, 16, 17, 18	psrmulfval 21980	. . 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 5151	. . . . . 6 ⊢ (𝑏 = (1o × {𝑘}) → (𝑑 ∘r ≤ 𝑏 ↔ 𝑑 ∘r ≤ (1o × {𝑘})))
21	20	rabbidv 3440	. . . . 5 ⊢ (𝑏 = (1o × {𝑘}) → {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ 𝑏} = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})})
22		fvoveq1 7453	. . . . . 6 ⊢ (𝑏 = (1o × {𝑘}) → (𝐺‘(𝑏 ∘f − 𝑐)) = (𝐺‘((1o × {𝑘}) ∘f − 𝑐)))
23	22	oveq2d 7446	. . . . 5 ⊢ (𝑏 = (1o × {𝑘}) → ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘f − 𝑐))) = ((𝐹‘𝑐) · (𝐺‘((1o × {𝑘}) ∘f − 𝑐))))
24	21, 23	mpteq12dv 5238	. . . 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 7446	. . 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 7148	. 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 22263	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring)
28	11, 14	ringcl 20267	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
29	27, 28	syl3an1 1162	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
30		eqid 2734	. . . 4 ⊢ (coe1‘(𝐹 ∙ 𝐺)) = (coe1‘(𝐹 ∙ 𝐺))
31		eqid 2734	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↦ (1o × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1o × {𝑘}))
32	30, 11, 10, 31	coe1fval3 22225	. . 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 2734	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
35		eqid 2734	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
36		simpl1 1190	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
37		ringcmn 20295	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
38	36, 37	syl 17	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
39		fzfi 14009	. . . . . 6 ⊢ (0...𝑘) ∈ Fin
40	39	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
41		simpll1 1211	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42		simpll2 1212	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹 ∈ 𝐵)
43		eqid 2734	. . . . . . . . . 10 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
44	43, 11, 10, 34	coe1f2 22226	. . . . . . . . 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 13656	. . . . . . . . 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 7104	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅))
49		simpll3 1213	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺 ∈ 𝐵)
50		eqid 2734	. . . . . . . . . 10 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
51	50, 11, 10, 34	coe1f2 22226	. . . . . . . . 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 13592	. . . . . . . . 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 7104	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅))
56	34, 13	ringcl 20267	. . . . . . 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 1370	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅))
58	57	fmpttd 7134	. . . . 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 3500	. . . . . . . . 9 ⊢ (0...𝑘) ∈ V
60	59	mptex 7242	. . . . . . . 8 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V
61		funmpt 6605	. . . . . . . 8 ⊢ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
62		fvex 6919	. . . . . . . 8 ⊢ (0g‘𝑅) ∈ V
63	60, 61, 62	3pm3.2i 1338	. . . . . . 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 8200	. . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
66	65	dmmptss 6262	. . . . . . . . 9 ⊢ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ⊆ (0...𝑘)
67	64, 66	sstri 4004	. . . . . . . 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 9417	. . . . . . 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 692	. . . . . 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 2734	. . . . . . 7 ⊢ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} = {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}
73	72	coe1mul2lem2 22286	. . . . . 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 19948	. . . 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 5150	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑑 ∘r ≤ (1o × {𝑘}) ↔ 𝑐 ∘r ≤ (1o × {𝑘})))
77	76	elrab 3694	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})} ↔ (𝑐 ∈ (ℕ0 ↑m 1o) ∧ 𝑐 ∘r ≤ (1o × {𝑘})))
78	77	simprbi 496	. . . . . . . . 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 3689	. . . . . . . . . 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 22285	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑐 ∈ (ℕ0 ↑m 1o)) → (𝑐 ∘r ≤ (1o × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
84	80, 82, 83	syl2anc 584	. . . . . . . 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 2735	. . . . . . 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 2735	. . . . . . 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 6906	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐹)‘𝑥) = ((coe1‘𝐹)‘(𝑐‘∅)))
89		oveq2 7438	. . . . . . . . 9 ⊢ (𝑥 = (𝑐‘∅) → (𝑘 − 𝑥) = (𝑘 − (𝑐‘∅)))
90	89	fveq2d 6910	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) = ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))
91	88, 90	oveq12d 7448	. . . . . . 7 ⊢ (𝑥 = (𝑐‘∅) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) = (((coe1‘𝐹)‘(𝑐‘∅)) · ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))))
92	85, 86, 87, 91	fmptco 7148	. . . . . 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 1212	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝐹 ∈ 𝐵)
94	43	fvcoe1 22224	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑐 ∈ (ℕ0 ↑m 1o)) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅)))
95	93, 82, 94	syl2anc 584	. . . . . . . 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 8511	. . . . . . . . . . . . . 14 ⊢ 1o = {∅}
97		0ex 5312	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
98	96, 2, 97	mapsnconst 8930	. . . . . . . . . . . . 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 7446	. . . . . . . . . . 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 3481	. . . . . . . . . . . . 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 6921	. . . . . . . . . . . 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 7726	. . . . . . . . . . 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 2774	. . . . . . . . . 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 6910	. . . . . . . . 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 1213	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑m 1o) ∣ 𝑑 ∘r ≤ (1o × {𝑘})}) → 𝐺 ∈ 𝐵)
109		fznn0sub 13592	. . . . . . . . . . 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 22223	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1o × {(𝑘 − (𝑐‘∅))})))
112	108, 110, 111	syl2anc 584	. . . . . . . . 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 2777	. . . . . . . 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 7448	. . . . . . 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 5247	. . . . . 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 2777	. . . . 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 7446	. . . 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 2774	. . 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 5247	. 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 2784	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 1086   = wceq 1536   ∈ wcel 2105  {crab 3432  Vcvv 3477   ⊆ wss 3962  ∅c0 4338  {csn 4630   class class class wbr 5147   ↦ cmpt 5230   × cxp 5686  dom cdm 5688   ∘ ccom 5692  Oncon0 6385  Fun wfun 6556  ⟶wf 6558  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430   ∘_f cof 7694   ∘_r cofr 7695   supp csupp 8183  1_oc1o 8497   ↑_m cmap 8864  Fincfn 8983   finSupp cfsupp 9398  0cc0 11152   ≤ cle 11293   − cmin 11489  ℕ₀cn0 12523  ...cfz 13543  Basecbs 17244  ._rcmulr 17298  0_gc0g 17485   Σ_g cgsu 17486  CMndccmn 19812  Ringcrg 20250   mPwSer cmps 21941  PwSer₁cps1 22191  coe₁cco1 22194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-mulg 19098  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-psr 21946  df-opsr 21950  df-psr1 22196  df-coe1 22199

This theorem is referenced by:  coe1mul  22288