Theorem imasring 20142

Description: The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
imasring.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasring.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasring.p	⊢ + = (+g‘𝑅)
imasring.t	⊢ · = (.r‘𝑅)
imasring.o	⊢ 1 = (1r‘𝑅)
imasring.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasring.e1	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasring.e2	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
imasring.r	⊢ (𝜑 → 𝑅 ∈ Ring)

Assertion

Ref	Expression
imasring	⊢ (𝜑 → (𝑈 ∈ Ring ∧ (𝐹‘ 1 ) = (1r‘𝑈)))

Distinct variable groups:   𝑞,𝑝, +   𝑎,𝑏,𝑝,𝑞,𝜑   𝑈,𝑎,𝑏,𝑝,𝑞   1 ,𝑝,𝑞   𝐵,𝑝,𝑞   𝐹,𝑎,𝑏,𝑝,𝑞   𝑅,𝑝,𝑞   𝑉,𝑎,𝑏,𝑝,𝑞   · ,𝑝,𝑞

Allowed substitution hints:   𝐵(𝑎,𝑏)   + (𝑎,𝑏)   𝑅(𝑎,𝑏)   · (𝑎,𝑏)   1 (𝑎,𝑏)

Proof of Theorem imasring

Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasring.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasring.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasring.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
4		imasring.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
5	1, 2, 3, 4	imasbas 17457	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2733	. . 3 ⊢ (𝜑 → (+g‘𝑈) = (+g‘𝑈))
7		eqidd 2733	. . 3 ⊢ (𝜑 → (.r‘𝑈) = (.r‘𝑈))
8		imasring.p	. . . . . 6 ⊢ + = (+g‘𝑅)
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → + = (+g‘𝑅))
10		imasring.e1	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
11		ringgrp 20060	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
12	4, 11	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
13		eqid 2732	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
14	1, 2, 9, 3, 10, 12, 13	imasgrp 18938	. . . 4 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈)))
15	14	simpld 495	. . 3 ⊢ (𝜑 → 𝑈 ∈ Grp)
16		imasring.e2	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
17		imasring.t	. . . . 5 ⊢ · = (.r‘𝑅)
18		eqid 2732	. . . . 5 ⊢ (.r‘𝑈) = (.r‘𝑈)
19	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑅 ∈ Ring)
20		simprl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉)
21	2	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
22	20, 21	eleqtrd 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ (Base‘𝑅))
23		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
24	23, 21	eleqtrd 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ (Base‘𝑅))
25		eqid 2732	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
26	25, 17	ringcl 20072	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
27	19, 22, 24, 26	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
28	27, 21	eleqtrrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
29	28	caovclg 7598	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
30	3, 16, 1, 2, 4, 17, 18, 29	imasmulf 17481	. . . 4 ⊢ (𝜑 → (.r‘𝑈):(𝐵 × 𝐵)⟶𝐵)
31		fovcdm 7576	. . . 4 ⊢ (((.r‘𝑈):(𝐵 × 𝐵)⟶𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(.r‘𝑈)𝑣) ∈ 𝐵)
32	30, 31	syl3an1 1163	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(.r‘𝑈)𝑣) ∈ 𝐵)
33		forn 6808	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
34	3, 33	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵)
35	34	eleq2d 2819	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
36	34	eleq2d 2819	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
37	34	eleq2d 2819	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
38	35, 36, 37	3anbi123d 1436	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
39		fofn 6807	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
40	3, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
41		fvelrnb 6952	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
42		fvelrnb 6952	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
43		fvelrnb 6952	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
44	41, 42, 43	3anbi123d 1436	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)))
45	40, 44	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)))
46	38, 45	bitr3d 280	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)))
47		3reeanv 3227	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
48	46, 47	bitr4di 288	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤)))
49	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Ring)
50		simp2 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
51	2	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
52	50, 51	eleqtrd 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
53	52	3adant3r3 1184	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
54		simp3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
55	54, 51	eleqtrd 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅))
56	55	3adant3r3 1184	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
57		simpr3 1196	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
58	2	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
59	57, 58	eleqtrd 2835	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅))
60	25, 17	ringass 20075	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
61	49, 53, 56, 59, 60	syl13anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
62	61	fveq2d 6895	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
63		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
64	28	caovclg 7598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
65	64	3adantr3 1171	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
66	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
67	63, 65, 57, 66	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
68		simpr1 1194	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
69	28	caovclg 7598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
70	69	3adantr1 1169	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
71	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
72	63, 68, 70, 71	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
73	62, 67, 72	3eqtr4d 2782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧))))
74	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
75	74	3adant3r3 1184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
76	75	oveq1d 7423	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)))
77	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
78	77	3adant3r1 1182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
79	78	oveq2d 7424	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) = ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧))))
80	73, 76, 79	3eqtr4d 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))))
81		simp1 1136	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
82		simp2 1137	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
83	81, 82	oveq12d 7426	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝑢(.r‘𝑈)𝑣))
84		simp3 1138	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
85	83, 84	oveq12d 7426	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤))
86	82, 84	oveq12d 7426	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝑣(.r‘𝑈)𝑤))
87	81, 86	oveq12d 7426	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))
88	85, 87	eqeq12d 2748	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) ↔ ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))))
89	80, 88	syl5ibcom 244	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))))
90	89	3exp2 1354	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))))))
91	90	imp32 419	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))))
92	91	rexlimdv 3153	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))))
93	92	rexlimdvva 3211	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))))
94	48, 93	sylbid 239	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤))))
95	94	imp 407	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(.r‘𝑈)𝑣)(.r‘𝑈)𝑤) = (𝑢(.r‘𝑈)(𝑣(.r‘𝑈)𝑤)))
96	25, 8, 17	ringdi 20080	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
97	49, 53, 56, 59, 96	syl13anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
98	97	fveq2d 6895	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
99	25, 8	ringacl 20094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
100	19, 22, 24, 99	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
101	100, 21	eleqtrrd 2836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
102	101	caovclg 7598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
103	102	3adantr1 1169	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
104	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
105	63, 68, 103, 104	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
106	28	caovclg 7598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
107	106	3adantr2 1170	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
108		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑈) = (+g‘𝑈)
109	3, 10, 1, 2, 4, 8, 108	imasaddval 17477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ (𝑥 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(+g‘𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
110	63, 65, 107, 109	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(+g‘𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
111	98, 105, 110	3eqtr4d 2782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g‘𝑈)(𝐹‘(𝑥 · 𝑧))))
112	3, 10, 1, 2, 4, 8, 108	imasaddval 17477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
113	112	3adant3r1 1182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
114	113	oveq2d 7424	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧))) = ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))))
115	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
116	115	3adant3r2 1183	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
117	75, 116	oveq12d 7426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(+g‘𝑈)((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g‘𝑈)(𝐹‘(𝑥 · 𝑧))))
118	111, 114, 117	3eqtr4d 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧))) = (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(+g‘𝑈)((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))))
119	82, 84	oveq12d 7426	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝑣(+g‘𝑈)𝑤))
120	81, 119	oveq12d 7426	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧))) = (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)))
121	81, 84	oveq12d 7426	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝑢(.r‘𝑈)𝑤))
122	83, 121	oveq12d 7426	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(+g‘𝑈)((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))
123	120, 122	eqeq12d 2748	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(.r‘𝑈)((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧))) = (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦))(+g‘𝑈)((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))) ↔ (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))))
124	118, 123	syl5ibcom 244	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))))
125	124	3exp2 1354	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))))))
126	125	imp32 419	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))))
127	126	rexlimdv 3153	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))))
128	127	rexlimdvva 3211	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))))
129	48, 128	sylbid 239	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤))))
130	129	imp 407	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(.r‘𝑈)(𝑣(+g‘𝑈)𝑤)) = ((𝑢(.r‘𝑈)𝑣)(+g‘𝑈)(𝑢(.r‘𝑈)𝑤)))
131	25, 8, 17	ringdir 20081	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
132	49, 53, 56, 59, 131	syl13anc 1372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
133	132	fveq2d 6895	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
134	101	caovclg 7598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
135	134	3adantr3 1171	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
136	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
137	63, 135, 57, 136	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
138	3, 10, 1, 2, 4, 8, 108	imasaddval 17477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 · 𝑧) ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑧))(+g‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
139	63, 107, 70, 138	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑧))(+g‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
140	133, 137, 139	3eqtr4d 2782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 · 𝑧))(+g‘𝑈)(𝐹‘(𝑦 · 𝑧))))
141	3, 10, 1, 2, 4, 8, 108	imasaddval 17477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
142	141	3adant3r3 1184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
143	142	oveq1d 7423	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧)))
144	116, 78	oveq12d 7426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))(+g‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) = ((𝐹‘(𝑥 · 𝑧))(+g‘𝑈)(𝐹‘(𝑦 · 𝑧))))
145	140, 143, 144	3eqtr4d 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))(+g‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))))
146	81, 82	oveq12d 7426	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝑢(+g‘𝑈)𝑣))
147	146, 84	oveq12d 7426	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤))
148	121, 86	oveq12d 7426	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))(+g‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))
149	147, 148	eqeq12d 2748	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧))(+g‘𝑈)((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) ↔ ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))))
150	145, 149	syl5ibcom 244	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))))
151	150	3exp2 1354	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))))))
152	151	imp32 419	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑧 ∈ 𝑉 → (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))))
153	152	rexlimdv 3153	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))))
154	153	rexlimdvva 3211	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))))
155	48, 154	sylbid 239	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤))))
156	155	imp 407	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝑈)𝑣)(.r‘𝑈)𝑤) = ((𝑢(.r‘𝑈)𝑤)(+g‘𝑈)(𝑣(.r‘𝑈)𝑤)))
157		fof 6805	. . . . 5 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
158	3, 157	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
159		imasring.o	. . . . . . 7 ⊢ 1 = (1r‘𝑅)
160	25, 159	ringidcl 20082	. . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
161	4, 160	syl 17	. . . . 5 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
162	161, 2	eleqtrrd 2836	. . . 4 ⊢ (𝜑 → 1 ∈ 𝑉)
163	158, 162	ffvelcdmd 7087	. . 3 ⊢ (𝜑 → (𝐹‘ 1 ) ∈ 𝐵)
164	40, 41	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
165	35, 164	bitr3d 280	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
166		simpl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝜑)
167	162	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 1 ∈ 𝑉)
168		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
169	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 1 · 𝑥)))
170	166, 167, 168, 169	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 1 · 𝑥)))
171	2	eleq2d 2819	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
172	171	biimpa 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
173	25, 17, 159	ringlidm 20085	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ( 1 · 𝑥) = 𝑥)
174	4, 172, 173	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 1 · 𝑥) = 𝑥)
175	174	fveq2d 6895	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 1 · 𝑥)) = (𝐹‘𝑥))
176	170, 175	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥))
177		oveq2 7416	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 1 )(.r‘𝑈)(𝐹‘𝑥)) = ((𝐹‘ 1 )(.r‘𝑈)𝑢))
178		id 22	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → (𝐹‘𝑥) = 𝑢)
179	177, 178	eqeq12d 2748	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘ 1 )(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ ((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢))
180	176, 179	syl5ibcom 244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢))
181	180	rexlimdva 3155	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → ((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢))
182	165, 181	sylbid 239	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → ((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢))
183	182	imp 407	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢)
184	3, 16, 1, 2, 4, 17, 18	imasmulval 17480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 1 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘(𝑥 · 1 )))
185	167, 184	mpd3an3 1462	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘(𝑥 · 1 )))
186	25, 17, 159	ringridm 20086	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 · 1 ) = 𝑥)
187	4, 172, 186	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 · 1 ) = 𝑥)
188	187	fveq2d 6895	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 · 1 )) = (𝐹‘𝑥))
189	185, 188	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘𝑥))
190		oveq1 7415	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝑢(.r‘𝑈)(𝐹‘ 1 )))
191	190, 178	eqeq12d 2748	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘𝑥) ↔ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢))
192	189, 191	syl5ibcom 244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢))
193	192	rexlimdva 3155	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢))
194	165, 193	sylbid 239	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢))
195	194	imp 407	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)
196	5, 6, 7, 15, 32, 95, 130, 156, 163, 183, 195	isringd 20104	. 2 ⊢ (𝜑 → 𝑈 ∈ Ring)
197	163, 5	eleqtrd 2835	. . . 4 ⊢ (𝜑 → (𝐹‘ 1 ) ∈ (Base‘𝑈))
198	5	eleq2d 2819	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ 𝑢 ∈ (Base‘𝑈)))
199	182, 194	jcad 513	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → (((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)))
200	198, 199	sylbird 259	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ (Base‘𝑈) → (((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)))
201	200	ralrimiv 3145	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢))
202		eqid 2732	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
203		eqid 2732	. . . . . 6 ⊢ (1r‘𝑈) = (1r‘𝑈)
204	202, 18, 203	isringid 20087	. . . . 5 ⊢ (𝑈 ∈ Ring → (((𝐹‘ 1 ) ∈ (Base‘𝑈) ∧ ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) ↔ (1r‘𝑈) = (𝐹‘ 1 )))
205	196, 204	syl 17	. . . 4 ⊢ (𝜑 → (((𝐹‘ 1 ) ∈ (Base‘𝑈) ∧ ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)) ↔ (1r‘𝑈) = (𝐹‘ 1 )))
206	197, 201, 205	mpbi2and 710	. . 3 ⊢ (𝜑 → (1r‘𝑈) = (𝐹‘ 1 ))
207	206	eqcomd 2738	. 2 ⊢ (𝜑 → (𝐹‘ 1 ) = (1r‘𝑈))
208	196, 207	jca 512	1 ⊢ (𝜑 → (𝑈 ∈ Ring ∧ (𝐹‘ 1 ) = (1r‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   × cxp 5674  ran crn 5677   Fn wfn 6538  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  +_gcplusg 17196  ._rcmulr 17197  0_gc0g 17384   “_s cimas 17449  Grpcgrp 18818  1_rcur 20003  Ringcrg 20055

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-fz 13484  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-0g 17386  df-imas 17453  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-mgp 19987  df-ur 20004  df-ring 20057

This theorem is referenced by:  imasringf1  20143  qusring2  20146