Theorem imasring 20218

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 17462	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2731	. . 3 ⊢ (𝜑 → (+g‘𝑈) = (+g‘𝑈))
7		eqidd 2731	. . 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 20132	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
12	4, 11	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
13		eqid 2730	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
14	1, 2, 9, 3, 10, 12, 13	imasgrp 18975	. . . 4 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈)))
15	14	simpld 493	. . 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 2730	. . . . 5 ⊢ (.r‘𝑈) = (.r‘𝑈)
19	4	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑅 ∈ Ring)
20		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉)
21	2	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
22	20, 21	eleqtrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ (Base‘𝑅))
23		simprr 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
24	23, 21	eleqtrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ (Base‘𝑅))
25		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
26	25, 17	ringcl 20144	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
27	19, 22, 24, 26	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
28	27, 21	eleqtrrd 2834	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
29	28	caovclg 7601	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
30	3, 16, 1, 2, 4, 17, 18, 29	imasmulf 17486	. . . 4 ⊢ (𝜑 → (.r‘𝑈):(𝐵 × 𝐵)⟶𝐵)
31		fovcdm 7579	. . . 4 ⊢ (((.r‘𝑈):(𝐵 × 𝐵)⟶𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(.r‘𝑈)𝑣) ∈ 𝐵)
32	30, 31	syl3an1 1161	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(.r‘𝑈)𝑣) ∈ 𝐵)
33		forn 6807	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
34	3, 33	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵)
35	34	eleq2d 2817	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
36	34	eleq2d 2817	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
37	34	eleq2d 2817	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
38	35, 36, 37	3anbi123d 1434	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
39		fofn 6806	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
40	3, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
41		fvelrnb 6951	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
42		fvelrnb 6951	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
43		fvelrnb 6951	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
44	41, 42, 43	3anbi123d 1434	. . . . . . . 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 3225	. . . . . 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 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Ring)
50		simp2 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
51	2	3ad2ant1 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
52	50, 51	eleqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
53	52	3adant3r3 1182	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
54		simp3 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
55	54, 51	eleqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅))
56	55	3adant3r3 1182	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
57		simpr3 1194	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
58	2	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
59	57, 58	eleqtrd 2833	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅))
60	25, 17	ringass 20147	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
62	61	fveq2d 6894	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
63		simpl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
64	28	caovclg 7601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
65	64	3adantr3 1169	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
66	3, 16, 1, 2, 4, 17, 18	imasmulval 17485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
67	63, 65, 57, 66	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
68		simpr1 1192	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
69	28	caovclg 7601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
70	69	3adantr1 1167	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
71	3, 16, 1, 2, 4, 17, 18	imasmulval 17485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
72	63, 68, 70, 71	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
73	62, 67, 72	3eqtr4d 2780	. . . . . . . . . . 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 17485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
75	74	3adant3r3 1182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
76	75	oveq1d 7426	. . . . . . . . . . 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 17485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
78	77	3adant3r1 1180	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
79	78	oveq2d 7427	. . . . . . . . . . 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 2780	. . . . . . . . . 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 1134	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
82		simp2 1135	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
83	81, 82	oveq12d 7429	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑦)) = (𝑢(.r‘𝑈)𝑣))
84		simp3 1136	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
85	83, 84	oveq12d 7429	. . . . . . . . . . 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 7429	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧)) = (𝑣(.r‘𝑈)𝑤))
87	81, 86	oveq12d 7429	. . . . . . . . . . 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 2746	. . . . . . . . . 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 1352	. . . . . . . 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 417	. . . . . . 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 3151	. . . . . 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 3209	. . . . 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 405	. . 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 20152	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
98	97	fveq2d 6894	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
99	25, 8	ringacl 20166	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
100	19, 22, 24, 99	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
101	100, 21	eleqtrrd 2834	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
102	101	caovclg 7601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
103	102	3adantr1 1167	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
104	3, 16, 1, 2, 4, 17, 18	imasmulval 17485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
105	63, 68, 103, 104	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
106	28	caovclg 7601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
107	106	3adantr2 1168	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
108		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑈) = (+g‘𝑈)
109	3, 10, 1, 2, 4, 8, 108	imasaddval 17482	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 17482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
113	112	3adant3r1 1180	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
114	113	oveq2d 7427	. . . . . . . . . . 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 17485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
116	115	3adant3r2 1181	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
117	75, 116	oveq12d 7429	. . . . . . . . . . 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 2780	. . . . . . . . . 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 7429	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+g‘𝑈)(𝐹‘𝑧)) = (𝑣(+g‘𝑈)𝑤))
120	81, 119	oveq12d 7429	. . . . . . . . . . 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 7429	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘𝑧)) = (𝑢(.r‘𝑈)𝑤))
122	83, 121	oveq12d 7429	. . . . . . . . . . 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 2746	. . . . . . . . . 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 1352	. . . . . . . 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 417	. . . . . . 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 3151	. . . . . 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 3209	. . . . 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 405	. . 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 20153	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
133	132	fveq2d 6894	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
134	101	caovclg 7601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
135	134	3adantr3 1169	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
136	3, 16, 1, 2, 4, 17, 18	imasmulval 17485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(.r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
137	63, 135, 57, 136	syl3anc 1369	. . . . . . . . . . . 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 17482	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 17482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
142	141	3adant3r3 1182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
143	142	oveq1d 7426	. . . . . . . . . . 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 7429	. . . . . . . . . . 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 2780	. . . . . . . . . 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 7429	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) = (𝑢(+g‘𝑈)𝑣))
147	146, 84	oveq12d 7429	. . . . . . . . . . 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 7429	. . . . . . . . . . 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 2746	. . . . . . . . . 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 1352	. . . . . . . 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 417	. . . . . . 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 3151	. . . . . 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 3209	. . . . 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 405	. . 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 6804	. . . . 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 20154	. . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
161	4, 160	syl 17	. . . . 5 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
162	161, 2	eleqtrrd 2834	. . . 4 ⊢ (𝜑 → 1 ∈ 𝑉)
163	158, 162	ffvelcdmd 7086	. . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝜑)
167	162	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 1 ∈ 𝑉)
168		simpr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
169	3, 16, 1, 2, 4, 17, 18	imasmulval 17485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 1 · 𝑥)))
170	166, 167, 168, 169	syl3anc 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 1 · 𝑥)))
171	2	eleq2d 2817	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
172	171	biimpa 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
173	25, 17, 159	ringlidm 20157	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ( 1 · 𝑥) = 𝑥)
174	4, 172, 173	syl2an2r 681	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 1 · 𝑥) = 𝑥)
175	174	fveq2d 6894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 1 · 𝑥)) = (𝐹‘𝑥))
176	170, 175	eqtrd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(.r‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥))
177		oveq2 7419	. . . . . . . 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 2746	. . . . . . 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 3153	. . . . 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 405	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢)
184	3, 16, 1, 2, 4, 17, 18	imasmulval 17485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 1 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘(𝑥 · 1 )))
185	167, 184	mpd3an3 1460	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘(𝑥 · 1 )))
186	25, 17, 159	ringridm 20158	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 · 1 ) = 𝑥)
187	4, 172, 186	syl2an2r 681	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 · 1 ) = 𝑥)
188	187	fveq2d 6894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 · 1 )) = (𝐹‘𝑥))
189	185, 188	eqtrd 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝐹‘𝑥))
190		oveq1 7418	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘𝑥)(.r‘𝑈)(𝐹‘ 1 )) = (𝑢(.r‘𝑈)(𝐹‘ 1 )))
191	190, 178	eqeq12d 2746	. . . . . . 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 3153	. . . . 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 405	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢)
196	5, 6, 7, 15, 32, 95, 130, 156, 163, 183, 195	isringd 20179	. 2 ⊢ (𝜑 → 𝑈 ∈ Ring)
197	163, 5	eleqtrd 2833	. . . 4 ⊢ (𝜑 → (𝐹‘ 1 ) ∈ (Base‘𝑈))
198	5	eleq2d 2817	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ 𝑢 ∈ (Base‘𝑈)))
199	182, 194	jcad 511	. . . . . 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 3143	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1 )(.r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(.r‘𝑈)(𝐹‘ 1 )) = 𝑢))
202		eqid 2730	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
203		eqid 2730	. . . . . 6 ⊢ (1r‘𝑈) = (1r‘𝑈)
204	202, 18, 203	isringid 20159	. . . . 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 708	. . 3 ⊢ (𝜑 → (1r‘𝑈) = (𝐹‘ 1 ))
207	206	eqcomd 2736	. 2 ⊢ (𝜑 → (𝐹‘ 1 ) = (1r‘𝑈))
208	196, 207	jca 510	1 ⊢ (𝜑 → (𝑈 ∈ Ring ∧ (𝐹‘ 1 ) = (1r‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068   × cxp 5673  ran crn 5676   Fn wfn 6537  ⟶wf 6538  –onto→wfo 6540  ‘cfv 6542  (class class class)co 7411  Basecbs 17148  +_gcplusg 17201  ._rcmulr 17202  0_gc0g 17389   “_s cimas 17454  Grpcgrp 18855  1_rcur 20075  Ringcrg 20127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-0g 17391  df-imas 17458  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-mgp 20029  df-ur 20076  df-ring 20129

This theorem is referenced by:  imasringf1  20219  qusring2  20222  imasrhm  32741