Theorem coe1mul2 19460

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 6007	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (1𝑜 × {𝑘}):1𝑜⟶ℕ0)
2		nn0ex 11175	. . . . . 6 ⊢ ℕ0 ∈ V
3		1on 7454	. . . . . . 7 ⊢ 1𝑜 ∈ On
4	3	elexi 3186	. . . . . 6 ⊢ 1𝑜 ∈ V
5	2, 4	elmap 7772	. . . . 5 ⊢ ((1𝑜 × {𝑘}) ∈ (ℕ0 ↑𝑚 1𝑜) ↔ (1𝑜 × {𝑘}):1𝑜⟶ℕ0)
6	1, 5	sylibr 223	. . . 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 2611	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})))
9		eqid 2610	. . . 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 19381	. . . 4 ⊢ 𝐵 = (Base‘(1𝑜 mPwSer 𝑅))
13		coe1mul2.u	. . . 4 ⊢ · = (.r‘𝑅)
14		coe1mul2.t	. . . . 5 ⊢ ∙ = (.r‘𝑆)
15	10, 9, 14	psr1mulr 19415	. . . 4 ⊢ ∙ = (.r‘(1𝑜 mPwSer 𝑅))
16		psr1baslem 19376	. . . 4 ⊢ (ℕ0 ↑𝑚 1𝑜) = {𝑎 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ (◡𝑎 “ ℕ) ∈ Fin}
17		simp2 1055	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
18		simp3 1056	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
19	9, 12, 13, 15, 16, 17, 18	psrmulfval 19206	. . 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 4587	. . . . . 6 ⊢ (𝑏 = (1𝑜 × {𝑘}) → (𝑑 ∘𝑟 ≤ 𝑏 ↔ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})))
21	20	rabbidv 3164	. . . . 5 ⊢ (𝑏 = (1𝑜 × {𝑘}) → {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ 𝑏} = {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})})
22		oveq1 6556	. . . . . . 7 ⊢ (𝑏 = (1𝑜 × {𝑘}) → (𝑏 ∘𝑓 − 𝑐) = ((1𝑜 × {𝑘}) ∘𝑓 − 𝑐))
23	22	fveq2d 6107	. . . . . 6 ⊢ (𝑏 = (1𝑜 × {𝑘}) → (𝐺‘(𝑏 ∘𝑓 − 𝑐)) = (𝐺‘((1𝑜 × {𝑘}) ∘𝑓 − 𝑐)))
24	23	oveq2d 6565	. . . . 5 ⊢ (𝑏 = (1𝑜 × {𝑘}) → ((𝐹‘𝑐) · (𝐺‘(𝑏 ∘𝑓 − 𝑐))) = ((𝐹‘𝑐) · (𝐺‘((1𝑜 × {𝑘}) ∘𝑓 − 𝑐))))
25	21, 24	mpteq12dv 4663	. . . 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)))))
26	25	oveq2d 6565	. . 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))))))
27	7, 8, 19, 26	fmptco 6303	. 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)))))))
28	10	psr1ring 19438	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Ring)
29	11, 14	ringcl 18384	. . . 4 ⊢ ((𝑆 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
30	28, 29	syl3an1 1351	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) ∈ 𝐵)
31		eqid 2610	. . . 4 ⊢ (coe1‘(𝐹 ∙ 𝐺)) = (coe1‘(𝐹 ∙ 𝐺))
32		eqid 2610	. . . 4 ⊢ (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘})) = (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘}))
33	31, 11, 10, 32	coe1fval3 19399	. . 3 ⊢ ((𝐹 ∙ 𝐺) ∈ 𝐵 → (coe1‘(𝐹 ∙ 𝐺)) = ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘}))))
34	30, 33	syl 17	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (coe1‘(𝐹 ∙ 𝐺)) = ((𝐹 ∙ 𝐺) ∘ (𝑘 ∈ ℕ0 ↦ (1𝑜 × {𝑘}))))
35		eqid 2610	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
36		eqid 2610	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
37		simpl1 1057	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ Ring)
38		ringcmn 18404	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
39	37, 38	syl 17	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑅 ∈ CMnd)
40		fzfi 12633	. . . . . 6 ⊢ (0...𝑘) ∈ Fin
41	40	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
42		simpll1 1093	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
43		simpll2 1094	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐹 ∈ 𝐵)
44		eqid 2610	. . . . . . . . . 10 ⊢ (coe1‘𝐹) = (coe1‘𝐹)
45	44, 11, 10, 35	coe1f2 19400	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹):ℕ0⟶(Base‘𝑅))
46	43, 45	syl 17	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1‘𝐹):ℕ0⟶(Base‘𝑅))
47		elfznn0 12302	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑘) → 𝑥 ∈ ℕ0)
48	47	adantl 481	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝑥 ∈ ℕ0)
49	46, 48	ffvelrnd 6268	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅))
50		simpll3 1095	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → 𝐺 ∈ 𝐵)
51		eqid 2610	. . . . . . . . . 10 ⊢ (coe1‘𝐺) = (coe1‘𝐺)
52	51, 11, 10, 35	coe1f2 19400	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅))
53	50, 52	syl 17	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (coe1‘𝐺):ℕ0⟶(Base‘𝑅))
54		fznn0sub 12244	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...𝑘) → (𝑘 − 𝑥) ∈ ℕ0)
55	54	adantl 481	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (𝑘 − 𝑥) ∈ ℕ0)
56	53, 55	ffvelrnd 6268	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅))
57	35, 13	ringcl 18384	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐹)‘𝑥) ∈ (Base‘𝑅) ∧ ((coe1‘𝐺)‘(𝑘 − 𝑥)) ∈ (Base‘𝑅)) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅))
58	42, 49, 56, 57	syl3anc 1318	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑥 ∈ (0...𝑘)) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) ∈ (Base‘𝑅))
59		eqid 2610	. . . . . 6 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) = (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
60	58, 59	fmptd 6292	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))):(0...𝑘)⟶(Base‘𝑅))
61	40	elexi 3186	. . . . . . . . 9 ⊢ (0...𝑘) ∈ V
62	61	mptex 6390	. . . . . . . 8 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ∈ V
63		funmpt 5840	. . . . . . . 8 ⊢ Fun (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))))
64		fvex 6113	. . . . . . . 8 ⊢ (0g‘𝑅) ∈ V
65	62, 63, 64	3pm3.2i 1232	. . . . . . 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)
66		suppssdm 7195	. . . . . . . . 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))))
67	59	dmmptss 5548	. . . . . . . . 9 ⊢ dom (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) ⊆ (0...𝑘)
68	66, 67	sstri 3577	. . . . . . . 8 ⊢ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘)
69	40, 68	pm3.2i 470	. . . . . . 7 ⊢ ((0...𝑘) ∈ Fin ∧ ((𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) supp (0g‘𝑅)) ⊆ (0...𝑘))
70		suppssfifsupp 8173	. . . . . . 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))
71	65, 69, 70	mp2an 704	. . . . . 6 ⊢ (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) finSupp (0g‘𝑅)
72	71	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑥 ∈ (0...𝑘) ↦ (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥)))) finSupp (0g‘𝑅))
73		eqid 2610	. . . . . . 7 ⊢ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})} = {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}
74	73	coe1mul2lem2 19459	. . . . . 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))
75	74	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))
76	35, 36, 39, 41, 60, 72, 75	gsumf1o 18140	. . . 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‘∅)))))
77		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑐 → (𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘}) ↔ 𝑐 ∘𝑟 ≤ (1𝑜 × {𝑘})))
78	77	elrab 3331	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})} ↔ (𝑐 ∈ (ℕ0 ↑𝑚 1𝑜) ∧ 𝑐 ∘𝑟 ≤ (1𝑜 × {𝑘})))
79	78	simprbi 479	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})} → 𝑐 ∘𝑟 ≤ (1𝑜 × {𝑘}))
80	79	adantl 481	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐 ∘𝑟 ≤ (1𝑜 × {𝑘}))
81		simplr 788	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑘 ∈ ℕ0)
82		elrabi 3328	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})} → 𝑐 ∈ (ℕ0 ↑𝑚 1𝑜))
83	82	adantl 481	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐 ∈ (ℕ0 ↑𝑚 1𝑜))
84		coe1mul2lem1 19458	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑐 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝑐 ∘𝑟 ≤ (1𝑜 × {𝑘}) ↔ (𝑐‘∅) ∈ (0...𝑘)))
85	81, 83, 84	syl2anc 691	. . . . . . . 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)))
86	80, 85	mpbid 221	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ (0...𝑘))
87		eqidd 2611	. . . . . . 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‘∅)))
88		eqidd 2611	. . . . . . 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)))))
89		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐹)‘𝑥) = ((coe1‘𝐹)‘(𝑐‘∅)))
90		oveq2 6557	. . . . . . . . 9 ⊢ (𝑥 = (𝑐‘∅) → (𝑘 − 𝑥) = (𝑘 − (𝑐‘∅)))
91	90	fveq2d 6107	. . . . . . . 8 ⊢ (𝑥 = (𝑐‘∅) → ((coe1‘𝐺)‘(𝑘 − 𝑥)) = ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))))
92	89, 91	oveq12d 6567	. . . . . . 7 ⊢ (𝑥 = (𝑐‘∅) → (((coe1‘𝐹)‘𝑥) · ((coe1‘𝐺)‘(𝑘 − 𝑥))) = (((coe1‘𝐹)‘(𝑐‘∅)) · ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅)))))
93	86, 87, 88, 92	fmptco 6303	. . . . . 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‘∅))))))
94		simpll2 1094	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → 𝐹 ∈ 𝐵)
95	44	fvcoe1 19398	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑐 ∈ (ℕ0 ↑𝑚 1𝑜)) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅)))
96	94, 83, 95	syl2anc 691	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → (𝐹‘𝑐) = ((coe1‘𝐹)‘(𝑐‘∅)))
97		df1o2 7459	. . . . . . . . . . . . . 14 ⊢ 1𝑜 = {∅}
98		0ex 4718	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
99	97, 2, 98	mapsnconst 7789	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (ℕ0 ↑𝑚 1𝑜) → 𝑐 = (1𝑜 × {(𝑐‘∅)}))
100	83, 99	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑐 = (1𝑜 × {(𝑐‘∅)}))
101	100	oveq2d 6565	. . . . . . . . . . 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‘∅)})))
102	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → 1𝑜 ∈ On)
103		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑘 ∈ V
104	103	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → 𝑘 ∈ V)
105		fvex 6113	. . . . . . . . . . . . 13 ⊢ (𝑐‘∅) ∈ V
106	105	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑐‘∅) ∈ V)
107	102, 104, 106	ofc12 6820	. . . . . . . . . . 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‘∅))}))
108	101, 107	eqtrd 2644	. . . . . . . . . 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‘∅))}))
109	108	fveq2d 6107	. . . . . . . . 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‘∅))})))
110		simpll3 1095	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → 𝐺 ∈ 𝐵)
111		fznn0sub 12244	. . . . . . . . . . 11 ⊢ ((𝑐‘∅) ∈ (0...𝑘) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
112	86, 111	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑘 ∈ ℕ0) ∧ 𝑐 ∈ {𝑑 ∈ (ℕ0 ↑𝑚 1𝑜) ∣ 𝑑 ∘𝑟 ≤ (1𝑜 × {𝑘})}) → (𝑘 − (𝑐‘∅)) ∈ ℕ0)
113	51	coe1fv 19397	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝐵 ∧ (𝑘 − (𝑐‘∅)) ∈ ℕ0) → ((coe1‘𝐺)‘(𝑘 − (𝑐‘∅))) = (𝐺‘(1𝑜 × {(𝑘 − (𝑐‘∅))})))
114	110, 112, 113	syl2anc 691	. . . . . . . . 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‘∅))})))
115	109, 114	eqtr4d 2647	. . . . . . . 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‘∅))))
116	96, 115	oveq12d 6567	. . . . . . 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‘∅)))))
117	116	mpteq2dva 4672	. . . . . 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‘∅))))))
118	93, 117	eqtr4d 2647	. . . . 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)))))
119	118	oveq2d 6565	. . . 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))))))
120	76, 119	eqtrd 2644	. . 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))))))
121	120	mpteq2dva 4672	. 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)))))))
122	27, 34, 121	3eqtr4d 2654	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038   ∘ ccom 5042  Oncon0 5640  Fun wfun 5798  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ∘_𝑟 cofr 6794   supp csupp 7182  1_𝑜c1o 7440   ↑_𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  0cc0 9815   ≤ cle 9954   − cmin 10145  ℕ₀cn0 11169  ...cfz 12197  Basecbs 15695  ._rcmulr 15769  0_gc0g 15923   Σ_g cgsu 15924  CMndccmn 18016  Ringcrg 18370   mPwSer cmps 19172  PwSer₁cps1 19366  coe₁cco1 19369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-ple 15788  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-mulg 17364  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-psr 19177  df-opsr 19181  df-psr1 19371  df-coe1 19374

This theorem is referenced by:  coe1mul  19461