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Theorem imasrng 20142

Description: The image structure of a non-unital ring is a non-unital ring (imasring 20295 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 17531	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2737	. 2 ⊢ (𝜑 → (+_g‘𝑈) = (+_g‘𝑈))
7		eqidd 2737	. 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 20120	. . . . 5 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
12	4, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Abel)
13		eqid 2736	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
14	1, 2, 9, 3, 10, 12, 13	imasabl 19862	. . 3 ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘(0_g‘𝑅)) = (0_g‘𝑈)))
15	14	simpld 494	. 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 2736	. . . 4 ⊢ (._r‘𝑈) = (._r‘𝑈)
19	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑅 ∈ Rng)
20		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉)
21	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
22	20, 21	eleqtrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ (Base‘𝑅))
23		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
24	23, 21	eleqtrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ (Base‘𝑅))
25		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
26	25, 17	rngcl 20129	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
27	19, 22, 24, 26	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
28	27, 21	eleqtrrd 2838	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
29	28	caovclg 7604	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
30	3, 16, 1, 2, 4, 17, 18, 29	imasmulf 17555	. . 3 ⊢ (𝜑 → (._r‘𝑈):(𝐵 × 𝐵)⟶𝐵)
31	30	fovcld 7539	. 2 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(._r‘𝑈)𝑣) ∈ 𝐵)
32		forn 6798	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
33	3, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
34	33	eleq2d 2821	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
35	33	eleq2d 2821	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
36	33	eleq2d 2821	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
37	34, 35, 36	3anbi123d 1438	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
38		fofn 6797	. . . . . . 7 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
39		fvelrnb 6944	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
40		fvelrnb 6944	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
41		fvelrnb 6944	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
42	39, 40, 41	3anbi123d 1438	. . . . . . 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 281	. . . . 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 3218	. . . . 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 289	. . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Rng)
48		simp2 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
49	2	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
50	48, 49	eleqtrd 2837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
51	50	3adant3r3 1185	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
52		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
53	52, 49	eleqtrd 2837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅))
54	53	3adant3r3 1185	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
55		simpr3 1197	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
56	2	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
57	55, 56	eleqtrd 2837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅))
58	25, 17	rngass 20124	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
60	59	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
61		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
62	28	caovclg 7604	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
63	62	3adantr3 1172	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
64	3, 16, 1, 2, 4, 17, 18	imasmulval 17554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
65	61, 63, 55, 64	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
66		simpr1 1195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
67	28	caovclg 7604	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
68	67	3adantr1 1170	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
69	3, 16, 1, 2, 4, 17, 18	imasmulval 17554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
70	61, 66, 68, 69	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
71	60, 65, 70	3eqtr4d 2781	. . . . . . . . . 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 17554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
73	72	3adant3r3 1185	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
74	73	oveq1d 7425	. . . . . . . . . 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 17554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
76	75	3adant3r1 1183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
77	76	oveq2d 7426	. . . . . . . . . 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 2781	. . . . . . . . 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 1136	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
80		simp2 1137	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
81	79, 80	oveq12d 7428	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝑢(._r‘𝑈)𝑣))
82		simp3 1138	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
83	81, 82	oveq12d 7428	. . . . . . . . . 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 7428	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝑣(._r‘𝑈)𝑤))
85	79, 84	oveq12d 7428	. . . . . . . . . 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 2752	. . . . . . . . 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 245	. . . . . . . 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 1355	. . . . . . 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 418	. . . . . 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 3140	. . . . 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 3202	. . . 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 240	. . 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 406	. 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 20125	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
96	95	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
97	25, 8	rngacl 20127	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Rng ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
98	19, 22, 24, 97	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
99	98, 21	eleqtrrd 2838	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
100	99	caovclg 7604	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
101	100	3adantr1 1170	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
102	3, 16, 1, 2, 4, 17, 18	imasmulval 17554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
103	61, 66, 101, 102	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
104	28	caovclg 7604	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
105	104	3adantr2 1171	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
106		eqid 2736	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
107	3, 10, 1, 2, 4, 8, 106	imasaddval 17551	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 2781	. . . . . . . . . 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 17551	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
111	110	3adant3r1 1183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
112	111	oveq2d 7426	. . . . . . . . . 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 17554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
114	113	3adant3r2 1184	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
115	73, 114	oveq12d 7428	. . . . . . . . . 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 2781	. . . . . . . . 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 7428	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝑣(+_g‘𝑈)𝑤))
118	79, 117	oveq12d 7428	. . . . . . . . . 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 7428	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝑢(._r‘𝑈)𝑤))
120	81, 119	oveq12d 7428	. . . . . . . . . 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 2752	. . . . . . . . 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 245	. . . . . . . 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 1355	. . . . . . 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 418	. . . . . 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 3140	. . . . 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 3202	. . . 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 240	. . 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 406	. 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 20126	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
131	130	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
132	99	caovclg 7604	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
133	132	3adantr3 1172	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
134	3, 16, 1, 2, 4, 17, 18	imasmulval 17554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
135	61, 133, 55, 134	syl3anc 1373	. . . . . . . . . . 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 17551	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 2781	. . . . . . . . . 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 17551	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
140	139	3adant3r3 1185	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
141	140	oveq1d 7425	. . . . . . . . . 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 7428	. . . . . . . . . 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 2781	. . . . . . . . 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 7428	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝑢(+_g‘𝑈)𝑣))
145	144, 82	oveq12d 7428	. . . . . . . . . 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 7428	. . . . . . . . . 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 2752	. . . . . . . . 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 245	. . . . . . . 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 1355	. . . . . . 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 418	. . . . . 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 3140	. . . . 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 3202	. . . 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 240	. . 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 406	. 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 20138	1 ⊢ (𝜑 → 𝑈 ∈ Rng)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3061 ran crn 5660 Fn wfn 6531 –onto→wfo 6534 ‘cfv 6536 (class class class)co 7410 Basecbs 17233 +_gcplusg 17276 ._rcmulr 17277 0_gc0g 17458 “_s cimas 17523 Abelcabl 19767 Rngcrng 20117

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-0g 17460 df-imas 17527 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-grp 18924 df-minusg 18925 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118

This theorem is referenced by: imasrngf1 20143 qusrng 20145

W3C validator