imasrng - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > imasrng		GIF version

Theorem imasrng 13885

Description: The image structure of a non-unital ring is a non-unital ring (imasring 13993 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 13306	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
6		eqidd 2210	. 2 ⊢ (𝜑 → (+_g‘𝑈) = (+_g‘𝑈))
7		eqidd 2210	. 2 ⊢ (𝜑 → (._r‘𝑈) = (._r‘𝑈))
8		imasrng.p	. . . . 5 ⊢ + = (+_g‘𝑅)
9	8	a1i 9	. . . 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 13864	. . . . 5 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
12	4, 11	syl 14	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Abel)
13		eqid 2209	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
14	1, 2, 9, 3, 10, 12, 13	imasabl 13839	. . 3 ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘(0_g‘𝑅)) = (0_g‘𝑈)))
15	14	simpld 112	. 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 2209	. . . 4 ⊢ (._r‘𝑈) = (._r‘𝑈)
19	4	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑅 ∈ Rng)
20		simprl 529	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ 𝑉)
21	2	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
22	20, 21	eleqtrd 2288	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑢 ∈ (Base‘𝑅))
23		simprr 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ 𝑉)
24	23, 21	eleqtrd 2288	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → 𝑣 ∈ (Base‘𝑅))
25		eqid 2209	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
26	25, 17	rngcl 13873	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
27	19, 22, 24, 26	syl3anc 1252	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
28	27, 21	eleqtrrd 2289	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
29	28	caovclg 6129	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
30	3, 16, 1, 2, 4, 17, 18, 29	imasmulf 13321	. . 3 ⊢ (𝜑 → (._r‘𝑈):(𝐵 × 𝐵)⟶𝐵)
31	30	fovcld 6080	. 2 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(._r‘𝑈)𝑣) ∈ 𝐵)
32		forn 5527	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
33	3, 32	syl 14	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
34	33	eleq2d 2279	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵))
35	33	eleq2d 2279	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵))
36	33	eleq2d 2279	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵))
37	34, 35, 36	3anbi123d 1327	. . . . . 6 ⊢ (𝜑 → ((𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹) ↔ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
38		fofn 5526	. . . . . . 7 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉)
39		fvelrnb 5654	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐹‘𝑥) = 𝑢))
40		fvelrnb 5654	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑉 (𝐹‘𝑦) = 𝑣))
41		fvelrnb 5654	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑉 (𝐹‘𝑧) = 𝑤))
42	39, 40, 41	3anbi123d 1327	. . . . . . 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 17	. . . . . 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 190	. . . . 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 2682	. . . . 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 198	. . . 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 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Rng)
48		simp2 1003	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
49	2	3ad2ant1 1023	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
50	48, 49	eleqtrd 2288	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅))
51	50	3adant3r3 1219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅))
52		simp3 1004	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
53	52, 49	eleqtrd 2288	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅))
54	53	3adant3r3 1219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅))
55		simpr3 1010	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉)
56	2	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
57	55, 56	eleqtrd 2288	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅))
58	25, 17	rngass 13868	. . . . . . . . . . . . 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 1254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
60	59	fveq2d 5607	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
61		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝜑)
62	28	caovclg 6129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
63	62	3adantr3 1163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
64	3, 16, 1, 2, 4, 17, 18	imasmulval 13320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
65	61, 63, 55, 64	syl3anc 1252	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘(𝑥 · 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
66		simpr1 1008	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ 𝑉)
67	28	caovclg 6129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
68	67	3adantr1 1161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
69	3, 16, 1, 2, 4, 17, 18	imasmulval 13320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
70	61, 66, 68, 69	syl3anc 1252	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
71	60, 65, 70	3eqtr4d 2252	. . . . . . . . . 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 13320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
73	72	3adant3r3 1219	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 · 𝑦)))
74	73	oveq1d 5989	. . . . . . . . . 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 13320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
76	75	3adant3r1 1217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 · 𝑧)))
77	76	oveq2d 5990	. . . . . . . . . 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 2252	. . . . . . . . 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 1002	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑥) = 𝑢)
80		simp2 1003	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑦) = 𝑣)
81	79, 80	oveq12d 5992	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑦)) = (𝑢(._r‘𝑈)𝑣))
82		simp3 1004	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → (𝐹‘𝑧) = 𝑤)
83	81, 82	oveq12d 5992	. . . . . . . . . 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 5992	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = (𝑣(._r‘𝑈)𝑤))
85	79, 84	oveq12d 5992	. . . . . . . . . 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 2224	. . . . . . . . 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 155	. . . . . . . 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 1230	. . . . . . 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 257	. . . . . 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 2627	. . . . 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 2636	. . . 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 150	. . 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 124	. 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 13869	. . . . . . . . . . . . 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 1254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
96	95	fveq2d 5607	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
97	25, 8	rngacl 13871	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Rng ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
98	19, 22, 24, 97	syl3anc 1252	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
99	98, 21	eleqtrrd 2289	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝑉 ∧ 𝑣 ∈ 𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
100	99	caovclg 6129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
101	100	3adantr1 1161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
102	3, 16, 1, 2, 4, 17, 18	imasmulval 13320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
103	61, 66, 101, 102	syl3anc 1252	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
104	28	caovclg 6129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
105	104	3adantr2 1162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
106		eqid 2209	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
107	3, 10, 1, 2, 4, 8, 106	imasaddval 13317	. . . . . . . . . . . 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 1252	. . . . . . . . . . 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 2252	. . . . . . . . . 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 13317	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
111	110	3adant3r1 1217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑦 + 𝑧)))
112	111	oveq2d 5990	. . . . . . . . . 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 13320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
114	113	3adant3r2 1218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘(𝑥 · 𝑧)))
115	73, 114	oveq12d 5992	. . . . . . . . . 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 2252	. . . . . . . . 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 5992	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑦)(+_g‘𝑈)(𝐹‘𝑧)) = (𝑣(+_g‘𝑈)𝑤))
118	79, 117	oveq12d 5992	. . . . . . . . . 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 5992	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(._r‘𝑈)(𝐹‘𝑧)) = (𝑢(._r‘𝑈)𝑤))
120	81, 119	oveq12d 5992	. . . . . . . . . 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 2224	. . . . . . . . 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 155	. . . . . . . 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 1230	. . . . . . 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 257	. . . . . 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 2627	. . . . 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 2636	. . . 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 150	. . 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 124	. 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 13870	. . . . . . . . . . . . 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 1254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
131	130	fveq2d 5607	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
132	99	caovclg 6129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
133	132	3adantr3 1163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
134	3, 16, 1, 2, 4, 17, 18	imasmulval 13320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝐹‘(𝑥 + 𝑦))(._r‘𝑈)(𝐹‘𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
135	61, 133, 55, 134	syl3anc 1252	. . . . . . . . . . 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 13317	. . . . . . . . . . . 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 1252	. . . . . . . . . . 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 2252	. . . . . . . . . 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 13317	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
140	139	3adant3r3 1219	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦)))
141	140	oveq1d 5989	. . . . . . . . . 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 5992	. . . . . . . . . 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 2252	. . . . . . . . 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 5992	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑥) = 𝑢 ∧ (𝐹‘𝑦) = 𝑣 ∧ (𝐹‘𝑧) = 𝑤) → ((𝐹‘𝑥)(+_g‘𝑈)(𝐹‘𝑦)) = (𝑢(+_g‘𝑈)𝑣))
145	144, 82	oveq12d 5992	. . . . . . . . . 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 5992	. . . . . . . . . 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 2224	. . . . . . . . 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 155	. . . . . . . 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 1230	. . . . . . 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 257	. . . . . 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 2627	. . . . 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 2636	. . . 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 150	. . 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 124	. 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 13882	1 ⊢ (𝜑 → 𝑈 ∈ Rng)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 983 = wceq 1375 ∈ wcel 2180 ∃wrex 2489 ran crn 4697 Fn wfn 5289 –onto→wfo 5292 ‘cfv 5294 (class class class)co 5974 Basecbs 12998 +_gcplusg 13076 ._rcmulr 13077 0_gc0g 13255 “_s cimas 13298 Abelcabl 13788 Rngcrng 13861

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-addcom 8067 ax-addass 8069 ax-i2m1 8072 ax-0lt1 8073 ax-0id 8075 ax-rnegex 8076 ax-pre-ltirr 8079 ax-pre-lttrn 8081 ax-pre-ltadd 8083

This theorem depends on definitions: df-bi 117 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-pw 3631 df-sn 3652 df-pr 3653 df-tp 3654 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-id 4361 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-pnf 8151 df-mnf 8152 df-ltxr 8154 df-inn 9079 df-2 9137 df-3 9138 df-ndx 13001 df-slot 13002 df-base 13004 df-sets 13005 df-plusg 13089 df-mulr 13090 df-0g 13257 df-iimas 13301 df-mgm 13355 df-sgrp 13401 df-mnd 13416 df-grp 13502 df-minusg 13503 df-cmn 13789 df-abl 13790 df-mgp 13850 df-rng 13862

This theorem is referenced by: imasrngf1 13886 qusrng 13887

W3C validator