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Theorem imasring 20328

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 = (1_r‘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 ) = (1_r‘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 17558	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2737	. . 3 ⊢ (𝜑 → (+_g‘𝑈) = (+_g‘𝑈))
7		eqidd 2737	. . 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 20236	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
12	4, 11	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp)
13		eqid 2736	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
14	1, 2, 9, 3, 10, 12, 13	imasgrp 19075	. . . 4 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0_g‘𝑅)) = (0_g‘𝑈)))
15	14	simpld 494	. . 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 2736	. . . . 5 ⊢ (._r‘𝑈) = (._r‘𝑈)
19	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑅 ∈ Ring)
20		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉)
21	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
22	20, 21	eleqtrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ (Base‘𝑅))
23		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
24	23, 21	eleqtrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ (Base‘𝑅))
25		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
26	25, 17	ringcl 20248	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
27	19, 22, 24, 26	syl3anc 1372	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
28	27, 21	eleqtrrd 2843	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
29	28	caovclg 7626	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
30	3, 16, 1, 2, 4, 17, 18, 29	imasmulf 17582	. . . 4 ⊢ (𝜑 → (._r‘𝑈):(𝐵 × 𝐵)⟶𝐵)
31		fovcdm 7604	. . . 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 6822	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
34	3, 33	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵)
35	34	eleq2d 2826	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
36	34	eleq2d 2826	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
37	34	eleq2d 2826	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
38	35, 36, 37	3anbi123d 1437	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
39		fofn 6821	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
40	3, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
41		fvelrnb 6968	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
42		fvelrnb 6968	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
43		fvelrnb 6968	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
44	41, 42, 43	3anbi123d 1437	. . . . . . . 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 281	. . . . . 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 3229	. . . . . 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 289	. . . . 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 480	. . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
59	57, 58	eleqtrd 2842	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅))
60	25, 17	ringass 20251	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
62	61	fveq2d 6909	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
63		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
64	28	caovclg 7626	. . . . . . . . . . . . . 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 17581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
67	63, 65, 57, 66	syl3anc 1372	. . . . . . . . . . . 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 7626	. . . . . . . . . . . . . 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 17581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
72	63, 68, 70, 71	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
73	62, 67, 72	3eqtr4d 2786	. . . . . . . . . . 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 17581	. . . . . . . . . . . . 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 7447	. . . . . . . . . . 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 17581	. . . . . . . . . . . . 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 7448	. . . . . . . . . . 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 2786	. . . . . . . . . 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 7450	. . . . . . . . . . . 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 7450	. . . . . . . . . . 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 7450	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝑣(._r‘𝑈)𝑤))
87	81, 86	oveq12d 7450	. . . . . . . . . . 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 2752	. . . . . . . . . 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 245	. . . . . . . . 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 418	. . . . . . 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 3152	. . . . . 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 3212	. . . . 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 240	. . . 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 406	. . 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 20259	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
98	97	fveq2d 6909	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
99	25, 8	ringacl 20276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
100	19, 22, 24, 99	syl3anc 1372	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
101	100, 21	eleqtrrd 2843	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
102	101	caovclg 7626	. . . . . . . . . . . . . 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 17581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
105	63, 68, 103, 104	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
106	28	caovclg 7626	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
109	3, 10, 1, 2, 4, 8, 108	imasaddval 17578	. . . . . . . . . . . . 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 1372	. . . . . . . . . . . 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 2786	. . . . . . . . . . 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 17578	. . . . . . . . . . . . 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 7448	. . . . . . . . . . 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 17581	. . . . . . . . . . . . 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 7450	. . . . . . . . . . 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 2786	. . . . . . . . . 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 7450	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝑣(+_g‘𝑈)𝑤))
120	81, 119	oveq12d 7450	. . . . . . . . . . 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 7450	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝑢(._r‘𝑈)𝑤))
122	83, 121	oveq12d 7450	. . . . . . . . . . 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 2752	. . . . . . . . . 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 245	. . . . . . . . 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 418	. . . . . . 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 3152	. . . . . 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 3212	. . . . 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 240	. . . 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 406	. . 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 20260	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
133	132	fveq2d 6909	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
134	101	caovclg 7626	. . . . . . . . . . . . . 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 17581	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
137	63, 135, 57, 136	syl3anc 1372	. . . . . . . . . . . 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 17578	. . . . . . . . . . . . 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 1372	. . . . . . . . . . . 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 2786	. . . . . . . . . . 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 17578	. . . . . . . . . . . . 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 7447	. . . . . . . . . . 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 7450	. . . . . . . . . . 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 2786	. . . . . . . . . 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 7450	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝑢(+_g‘𝑈)𝑣))
147	146, 84	oveq12d 7450	. . . . . . . . . . 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 7450	. . . . . . . . . . 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 2752	. . . . . . . . . 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 245	. . . . . . . . 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 418	. . . . . . 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 3152	. . . . . 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 3212	. . . . 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 240	. . . 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 406	. . 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 6819	. . . . 5 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵)
158	3, 157	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
159		imasring.o	. . . . . . 7 ⊢ 1 = (1_r‘𝑅)
160	25, 159	ringidcl 20263	. . . . . 6 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
161	4, 160	syl 17	. . . . 5 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
162	161, 2	eleqtrrd 2843	. . . 4 ⊢ (𝜑 → 1 ∈ 𝑉)
163	158, 162	ffvelcdmd 7104	. . 3 ⊢ (𝜑 → (𝐹‘ 1 ) ∈ 𝐵)
164	40, 41	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
165	35, 164	bitr3d 281	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
166		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝜑)
167	162	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 1 ∈ 𝑉)
168		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
169	3, 16, 1, 2, 4, 17, 18	imasmulval 17581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(._r‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 1 · 𝑥)))
170	166, 167, 168, 169	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(._r‘𝑈)(𝐹‘𝑥)) = (𝐹‘( 1 · 𝑥)))
171	2	eleq2d 2826	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅)))
172	171	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
173	25, 17, 159	ringlidm 20267	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ( 1 · 𝑥) = 𝑥)
174	4, 172, 173	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 1 · 𝑥) = 𝑥)
175	174	fveq2d 6909	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 1 · 𝑥)) = (𝐹‘𝑥))
176	170, 175	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘ 1 )(._r‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥))
177		oveq2 7440	. . . . . . . 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 2752	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘ 1 )(._r‘𝑈)(𝐹‘𝑥)) = (𝐹‘𝑥) ↔ ((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢))
180	176, 179	syl5ibcom 245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → ((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢))
181	180	rexlimdva 3154	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → ((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢))
182	165, 181	sylbid 240	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → ((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢))
183	182	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → ((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢)
184	3, 16, 1, 2, 4, 17, 18	imasmulval 17581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 1 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘ 1 )) = (𝐹‘(𝑥 · 1 )))
185	167, 184	mpd3an3 1463	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘ 1 )) = (𝐹‘(𝑥 · 1 )))
186	25, 17, 159	ringridm 20268	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 · 1 ) = 𝑥)
187	4, 172, 186	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 · 1 ) = 𝑥)
188	187	fveq2d 6909	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 · 1 )) = (𝐹‘𝑥))
189	185, 188	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘ 1 )) = (𝐹‘𝑥))
190		oveq1 7439	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑢 → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘ 1 )) = (𝑢(._r‘𝑈)(𝐹‘ 1 )))
191	190, 178	eqeq12d 2752	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑢 → (((𝐹‘𝑥)(._r‘𝑈)(𝐹‘ 1 )) = (𝐹‘𝑥) ↔ (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢))
192	189, 191	syl5ibcom 245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐹‘𝑥) = 𝑢 → (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢))
193	192	rexlimdva 3154	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 → (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢))
194	165, 193	sylbid 240	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢))
195	194	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢)
196	5, 6, 7, 15, 32, 95, 130, 156, 163, 183, 195	isringd 20289	. 2 ⊢ (𝜑 → 𝑈 ∈ Ring)
197	163, 5	eleqtrd 2842	. . . 4 ⊢ (𝜑 → (𝐹‘ 1 ) ∈ (Base‘𝑈))
198	5	eleq2d 2826	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↔ 𝑢 ∈ (Base‘𝑈)))
199	182, 194	jcad 512	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 → (((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢)))
200	198, 199	sylbird 260	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ (Base‘𝑈) → (((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢)))
201	200	ralrimiv 3144	. . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢))
202		eqid 2736	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
203		eqid 2736	. . . . . 6 ⊢ (1_r‘𝑈) = (1_r‘𝑈)
204	202, 18, 203	isringid 20269	. . . . 5 ⊢ (𝑈 ∈ Ring → (((𝐹‘ 1 ) ∈ (Base‘𝑈) ∧ ∀𝑢 ∈ (Base‘𝑈)(((𝐹‘ 1 )(._r‘𝑈)𝑢) = 𝑢 ∧ (𝑢(._r‘𝑈)(𝐹‘ 1 )) = 𝑢)) ↔ (1_r‘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)) ↔ (1_r‘𝑈) = (𝐹‘ 1 )))
206	197, 201, 205	mpbi2and 712	. . 3 ⊢ (𝜑 → (1_r‘𝑈) = (𝐹‘ 1 ))
207	206	eqcomd 2742	. 2 ⊢ (𝜑 → (𝐹‘ 1 ) = (1_r‘𝑈))
208	196, 207	jca 511	1 ⊢ (𝜑 → (𝑈 ∈ Ring ∧ (𝐹‘ 1 ) = (1_r‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 × cxp 5682 ran crn 5685 Fn wfn 6555 ⟶wf 6556 –onto→wfo 6558 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +_gcplusg 17298 ._rcmulr 17299 0_gc0g 17485 “_s cimas 17550 Grpcgrp 18952 1_rcur 20179 Ringcrg 20231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17487 df-imas 17554 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-mgp 20139 df-ur 20180 df-ring 20233

This theorem is referenced by: imasringf1 20329 qusring2 20332 imasrhm 33385

W3C validator