Theorem isring 12976

Description: The predicate "is a (unital) ring". Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
isring.b	⊢ 𝐵 = (Base‘𝑅)
isring.g	⊢ 𝐺 = (mulGrp‘𝑅)
isring.p	⊢ + = (+g‘𝑅)
isring.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
isring	⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥, + ,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥, · ,𝑦,𝑧

Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem isring

Dummy variables 𝑝 𝑏 𝑟 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5507	. . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2		isring.g	. . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅)
3	1, 2	eqtr4di 2226	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
4	3	eleq1d 2244	. . . 4 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ Mnd ↔ 𝐺 ∈ Mnd))
5		basfn 12485	. . . . . . 7 ⊢ Base Fn V
6		vex 2738	. . . . . . 7 ⊢ 𝑟 ∈ V
7		funfvex 5524	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
8	7	funfni 5308	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑟 ∈ V) → (Base‘𝑟) ∈ V)
9	5, 6, 8	mp2an 426	. . . . . 6 ⊢ (Base‘𝑟) ∈ V
10	9	a1i 9	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
11		fveq2 5507	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
12		isring.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
13	11, 12	eqtr4di 2226	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
14		plusgslid 12525	. . . . . . . . 9 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
15	14	slotex 12455	. . . . . . . 8 ⊢ (𝑟 ∈ V → (+g‘𝑟) ∈ V)
16	15	elv 2739	. . . . . . 7 ⊢ (+g‘𝑟) ∈ V
17	16	a1i 9	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (+g‘𝑟) ∈ V)
18		simpl 109	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → 𝑟 = 𝑅)
19	18	fveq2d 5511	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (+g‘𝑟) = (+g‘𝑅))
20		isring.p	. . . . . . 7 ⊢ + = (+g‘𝑅)
21	19, 20	eqtr4di 2226	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (+g‘𝑟) = + )
22		mulrslid 12542	. . . . . . . . . 10 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
23	22	slotex 12455	. . . . . . . . 9 ⊢ (𝑟 ∈ V → (.r‘𝑟) ∈ V)
24	23	elv 2739	. . . . . . . 8 ⊢ (.r‘𝑟) ∈ V
25	24	a1i 9	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r‘𝑟) ∈ V)
26		simpll 527	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → 𝑟 = 𝑅)
27	26	fveq2d 5511	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r‘𝑟) = (.r‘𝑅))
28		isring.t	. . . . . . . 8 ⊢ · = (.r‘𝑅)
29	27, 28	eqtr4di 2226	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r‘𝑟) = · )
30		simpllr 534	. . . . . . . 8 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑏 = 𝐵)
31		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑡 = · )
32		eqidd 2176	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑥 = 𝑥)
33		simplr 528	. . . . . . . . . . . . . 14 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑝 = + )
34	33	oveqd 5882	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑦𝑝𝑧) = (𝑦 + 𝑧))
35	31, 32, 34	oveq123d 5886	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡(𝑦𝑝𝑧)) = (𝑥 · (𝑦 + 𝑧)))
36	31	oveqd 5882	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡𝑦) = (𝑥 · 𝑦))
37	31	oveqd 5882	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡𝑧) = (𝑥 · 𝑧))
38	33, 36, 37	oveq123d 5886	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
39	35, 38	eqeq12d 2190	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ↔ (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))))
40	33	oveqd 5882	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
41		eqidd 2176	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑧 = 𝑧)
42	31, 40, 41	oveq123d 5886	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥 + 𝑦) · 𝑧))
43	31	oveqd 5882	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑦𝑡𝑧) = (𝑦 · 𝑧))
44	33, 37, 43	oveq123d 5886	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
45	42, 44	eqeq12d 2190	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) ↔ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
46	39, 45	anbi12d 473	. . . . . . . . . 10 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
47	30, 46	raleqbidv 2682	. . . . . . . . 9 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
48	30, 47	raleqbidv 2682	. . . . . . . 8 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
49	30, 48	raleqbidv 2682	. . . . . . 7 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
50	25, 29, 49	sbcied2 2998	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → ([(.r‘𝑟) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
51	17, 21, 50	sbcied2 2998	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → ([(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
52	10, 13, 51	sbcied2 2998	. . . 4 ⊢ (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
53	4, 52	anbi12d 473	. . 3 ⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)))) ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))))
54		df-ring 12974	. . 3 ⊢ Ring = {𝑟 ∈ Grp ∣ ((mulGrp‘𝑟) ∈ Mnd ∧ [(Base‘𝑟) / 𝑏][(+g‘𝑟) / 𝑝][(.r‘𝑟) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}
55	53, 54	elrab2 2894	. 2 ⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))))
56		3anass 982	. 2 ⊢ ((𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))) ↔ (𝑅 ∈ Grp ∧ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))))
57	55, 56	bitr4i 187	1 ⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2146  ∀wral 2453  Vcvv 2735  [wsbc 2960   Fn wfn 5203  ‘cfv 5208  (class class class)co 5865  Basecbs 12428  +_gcplusg 12492  ._rcmulr 12493  Mndcmnd 12682  Grpcgrp 12738  mulGrpcmgp 12925  Ringcrg 12972

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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-ov 5868  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12431  df-slot 12432  df-base 12434  df-plusg 12505  df-mulr 12506  df-ring 12974

This theorem is referenced by:  ringgrp  12977  ringmgp  12978  ringdilem  12988  ringpropd  13009  isringd  13012