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Theorem imasrng 20206

Description: The image structure of a non-unital ring is a non-unital ring (imasring 20358 analog). (Contributed by AV, 22-Feb-2025.)

**Hypotheses**
Ref	Expression
imasrng.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasrng.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasrng.p	⊢ + = (+_g‘𝑅)
imasrng.t	⊢ · = (._r‘𝑅)
imasrng.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasrng.e1	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasrng.e2	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
imasrng.r	⊢ (𝜑 → 𝑅 ∈ Rng)

**Assertion**
Ref	Expression
imasrng	⊢ (𝜑 → 𝑈 ∈ Rng)

Distinct variable groups: + ,𝑝,𝑞 𝜑,𝑎,𝑏,𝑝,𝑞 𝑈,𝑎,𝑏,𝑝,𝑞 𝐵,𝑎,𝑏,𝑝,𝑞 𝐹,𝑎,𝑏,𝑝,𝑞 𝑅,𝑎,𝑏,𝑝,𝑞 𝑉,𝑎,𝑏,𝑝,𝑞 · ,𝑝,𝑞 Allowed substitution hints: + (𝑎,𝑏) · (𝑎,𝑏)

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

Step	Hyp	Ref	Expression
1		imasrng.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasrng.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasrng.f	. . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
4		imasrng.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Rng)
5	1, 2, 3, 4	imasbas 17525	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2762	. 2 ⊢ (𝜑 → (+_g‘𝑈) = (+_g‘𝑈))
7		eqidd 2762	. 2 ⊢ (𝜑 → (._r‘𝑈) = (._r‘𝑈))
8		imasrng.p	. . . . 5 ⊢ + = (+_g‘𝑅)
9	8	a1i 11	. . . 4 ⊢ (𝜑 → + = (+_g‘𝑅))
10		imasrng.e1	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
11		rngabl 20184	. . . . 5 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
12	4, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Abel)
13		eqid 2761	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
14	1, 2, 9, 3, 10, 12, 13	imasabl 19899	. . 3 ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘(0_g‘𝑅)) = (0_g‘𝑈)))
15	14	simpld 498	. 2 ⊢ (𝜑 → 𝑈 ∈ Abel)
16		imasrng.e2	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
17		imasrng.t	. . . 4 ⊢ · = (._r‘𝑅)
18		eqid 2761	. . . 4 ⊢ (._r‘𝑈) = (._r‘𝑈)
19	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑅 ∈ Rng)
20		simprl 780	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉)
21	2	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
22	20, 21	eleqtrd 2863	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ (Base‘𝑅))
23		simprr 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
24	23, 21	eleqtrd 2863	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ (Base‘𝑅))
25		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
26	25, 17	rngcl 20193	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
27	19, 22, 24, 26	syl3anc 1389	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
28	27, 21	eleqtrrd 2864	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
29	28	caovclg 7584	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
30	3, 16, 1, 2, 4, 17, 18, 29	imasmulf 17549	. . 3 ⊢ (𝜑 → (._r‘𝑈):(𝐵 × 𝐵)⟶𝐵)
31	30	fovcld 7519	. 2 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(._r‘𝑈)𝑣) ∈ 𝐵)
32		forn 6777	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
33	3, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
34	33	eleq2d 2847	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
35	33	eleq2d 2847	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
36	33	eleq2d 2847	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
37	34, 35, 36	3anbi123d 1456	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
38		fofn 6776	. . . . . . 7 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
39		fvelrnb 6923	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
40		fvelrnb 6923	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
41		fvelrnb 6923	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
42	39, 40, 41	3anbi123d 1456	. . . . . . 7 ⊢ (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)))
43	3, 38, 42	3syl 18	. . . . . 6 ⊢ (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)))
44	37, 43	bitr3d 283	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢 ∧ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣 ∧ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤)))
45		3reeanv 3234	. . . . 5 ⊢ (∃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))
46	44, 45	bitr4di 291	. . . 4 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 ((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤)))
47	4	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Rng)
48		simp2 1149	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
49	2	3ad2ant1 1145	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
50	48, 49	eleqtrd 2863	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
51	50	3adant3r3 1197	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
52		simp3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
53	52, 49	eleqtrd 2863	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅))
54	53	3adant3r3 1197	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
55		simpr3 1209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
56	2	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
57	55, 56	eleqtrd 2863	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅))
58	25, 17	rngass 20188	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
59	47, 51, 54, 57, 58	syl13anc 1390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
60	59	fveq2d 6867	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
61		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
62	28	caovclg 7584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
63	62	3adantr3 1184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
64	3, 16, 1, 2, 4, 17, 18	imasmulval 17548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
65	61, 63, 55, 64	syl3anc 1389	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
66		simpr1 1207	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
67	28	caovclg 7584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
68	67	3adantr1 1182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
69	3, 16, 1, 2, 4, 17, 18	imasmulval 17548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
70	61, 66, 68, 69	syl3anc 1389	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
71	60, 65, 70	3eqtr4d 2806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))))
72	3, 16, 1, 2, 4, 17, 18	imasmulval 17548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
73	72	3adant3r3 1197	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
74	73	oveq1d 7407	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦))(._r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)))
75	3, 16, 1, 2, 4, 17, 18	imasmulval 17548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
76	75	3adant3r1 1195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
77	76	oveq2d 7408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧))) = ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))))
78	71, 74, 77	3eqtr4d 2806	. . . . . . . . 9 ⊢ ((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))))
79		simp1 1148	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
80		simp2 1149	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
81	79, 80	oveq12d 7410	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝑢(._r‘𝑈)𝑣))
82		simp3 1150	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
83	81, 82	oveq12d 7410	. . . . . . . . . 10 ⊢ (((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))
84	80, 82	oveq12d 7410	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝑣(._r‘𝑈)𝑤))
85	79, 84	oveq12d 7410	. . . . . . . . . 10 ⊢ (((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	83, 85	eqeq12d 2777	. . . . . . . . 9 ⊢ (((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))))
87	78, 86	syl5ibcom 247	. . . . . . . 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))))
88	87	3exp2 1367	. . . . . . 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)))))))
89	88	imp32 422	. . . . . 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)))))
90	89	rexlimdv 3160	. . . . 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))))
91	90	rexlimdvva 3218	. . . 4 ⊢ (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	46, 91	sylbid 242	. . 3 ⊢ (𝜑 → ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → ((𝑢(._r‘𝑈)𝑣)(._r‘𝑈)𝑤) = (𝑢(._r‘𝑈)(𝑣(._r‘𝑈)𝑤))))
93	92	imp 410	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(._r‘𝑈)𝑣)(._r‘𝑈)𝑤) = (𝑢(._r‘𝑈)(𝑣(._r‘𝑈)𝑤)))
94	25, 8, 17	rngdi 20189	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
95	47, 51, 54, 57, 94	syl13anc 1390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
96	95	fveq2d 6867	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
97	25, 8	rngacl 20191	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Rng ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
98	19, 22, 24, 97	syl3anc 1389	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
99	98, 21	eleqtrrd 2864	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
100	99	caovclg 7584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
101	100	3adantr1 1182	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
102	3, 16, 1, 2, 4, 17, 18	imasmulval 17548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
103	61, 66, 101, 102	syl3anc 1389	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
104	28	caovclg 7584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
105	104	3adantr2 1183	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
106		eqid 2761	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
107	3, 10, 1, 2, 4, 8, 106	imasaddval 17545	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ (𝑥 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(+_g‘𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
108	61, 63, 105, 107	syl3anc 1389	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(+_g‘𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
109	96, 103, 108	3eqtr4d 2806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+_g‘𝑈)(𝐹‘(𝑥 · 𝑧))))
110	3, 10, 1, 2, 4, 8, 106	imasaddval 17545	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
111	110	3adant3r1 1195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
112	111	oveq2d 7408	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧))) = ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))))
113	3, 16, 1, 2, 4, 17, 18	imasmulval 17548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
114	113	3adant3r2 1196	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
115	73, 114	oveq12d 7410	. . . . . . . . . 10 ⊢ ((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))))
116	109, 112, 115	3eqtr4d 2806	. . . . . . . . 9 ⊢ ((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))))
117	80, 82	oveq12d 7410	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝑣(+_g‘𝑈)𝑤))
118	79, 117	oveq12d 7410	. . . . . . . . . 10 ⊢ (((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)))
119	79, 82	oveq12d 7410	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝑢(._r‘𝑈)𝑤))
120	81, 119	oveq12d 7410	. . . . . . . . . 10 ⊢ (((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)))
121	118, 120	eqeq12d 2777	. . . . . . . . 9 ⊢ (((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))))
122	116, 121	syl5ibcom 247	. . . . . . . 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))))
123	122	3exp2 1367	. . . . . . 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)))))))
124	123	imp32 422	. . . . . 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)))))
125	124	rexlimdv 3160	. . . . 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))))
126	125	rexlimdvva 3218	. . . 4 ⊢ (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	46, 126	sylbid 242	. . 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))))
128	127	imp 410	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢(._r‘𝑈)(𝑣(+_g‘𝑈)𝑤)) = ((𝑢(._r‘𝑈)𝑣)(+_g‘𝑈)(𝑢(._r‘𝑈)𝑤)))
129	25, 8, 17	rngdir 20190	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
130	47, 51, 54, 57, 129	syl13anc 1390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
131	130	fveq2d 6867	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
132	99	caovclg 7584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
133	132	3adantr3 1184	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
134	3, 16, 1, 2, 4, 17, 18	imasmulval 17548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
135	61, 133, 55, 134	syl3anc 1389	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
136	3, 10, 1, 2, 4, 8, 106	imasaddval 17545	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 · 𝑧) ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑧))(+_g‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
137	61, 105, 68, 136	syl3anc 1389	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑧))(+_g‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
138	131, 135, 137	3eqtr4d 2806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 + 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 · 𝑧))(+_g‘𝑈)(𝐹‘(𝑦 · 𝑧))))
139	3, 10, 1, 2, 4, 8, 106	imasaddval 17545	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
140	139	3adant3r3 1197	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
141	140	oveq1d 7407	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦))(._r‘𝑈)(𝐹‘𝑧)) = ((𝐹‘(𝑥 + 𝑦))(._r‘𝑈)(𝐹‘𝑧)))
142	114, 76	oveq12d 7410	. . . . . . . . . 10 ⊢ ((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))))
143	138, 141, 142	3eqtr4d 2806	. . . . . . . . 9 ⊢ ((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))))
144	79, 80	oveq12d 7410	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝑢(+_g‘𝑈)𝑣))
145	144, 82	oveq12d 7410	. . . . . . . . . 10 ⊢ (((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))
146	119, 84	oveq12d 7410	. . . . . . . . . 10 ⊢ (((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)))
147	145, 146	eqeq12d 2777	. . . . . . . . 9 ⊢ (((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))))
148	143, 147	syl5ibcom 247	. . . . . . . 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))))
149	148	3exp2 1367	. . . . . . 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)))))))
150	149	imp32 422	. . . . . 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)))))
151	150	rexlimdv 3160	. . . . 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))))
152	151	rexlimdvva 3218	. . . 4 ⊢ (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	46, 152	sylbid 242	. . 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))))
154	153	imp 410	. 2 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝑈)𝑣)(._r‘𝑈)𝑤) = ((𝑢(._r‘𝑈)𝑤)(+_g‘𝑈)(𝑣(._r‘𝑈)𝑤)))
155	5, 6, 7, 15, 31, 93, 128, 154	isrngd 20202	1 ⊢ (𝜑 → 𝑈 ∈ Rng)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∃wrex 3085 ran crn 5646 Fn wfn 6512 –onto→wfo 6515 ‘cfv 6517 (class class class)co 7392 Basecbs 17228 +_gcplusg 17269 ._rcmulr 17270 0_gc0g 17451 “_s cimas 17517 Abelcabl 19804 Rngcrng 20181

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-inf 9386 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-fz 13510 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-0g 17453 df-imas 17521 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-grp 18961 df-minusg 18962 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182

This theorem is referenced by: imasrngf1 20207 qusrng 20209

W3C validator