Theorem isring 20177

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 6897	. . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2		isring.g	. . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅)
3	1, 2	eqtr4di 2786	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
4	3	eleq1d 2814	. . . 4 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ Mnd ↔ 𝐺 ∈ Mnd))
5		fvexd 6912	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
6		fveq2 6897	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
7		isring.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
8	6, 7	eqtr4di 2786	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
9		fvexd 6912	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (+g‘𝑟) ∈ V)
10		simpl 482	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → 𝑟 = 𝑅)
11	10	fveq2d 6901	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (+g‘𝑟) = (+g‘𝑅))
12		isring.p	. . . . . . 7 ⊢ + = (+g‘𝑅)
13	11, 12	eqtr4di 2786	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (+g‘𝑟) = + )
14		fvexd 6912	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r‘𝑟) ∈ V)
15		simpll 766	. . . . . . . . 9 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → 𝑟 = 𝑅)
16	15	fveq2d 6901	. . . . . . . 8 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r‘𝑟) = (.r‘𝑅))
17		isring.t	. . . . . . . 8 ⊢ · = (.r‘𝑅)
18	16, 17	eqtr4di 2786	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) → (.r‘𝑟) = · )
19		simpllr 775	. . . . . . . 8 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑏 = 𝐵)
20		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑡 = · )
21		eqidd 2729	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑥 = 𝑥)
22		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑝 = + )
23	22	oveqd 7437	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑦𝑝𝑧) = (𝑦 + 𝑧))
24	20, 21, 23	oveq123d 7441	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡(𝑦𝑝𝑧)) = (𝑥 · (𝑦 + 𝑧)))
25	20	oveqd 7437	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡𝑦) = (𝑥 · 𝑦))
26	20	oveqd 7437	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑡𝑧) = (𝑥 · 𝑧))
27	22, 25, 26	oveq123d 7441	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
28	24, 27	eqeq12d 2744	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ↔ (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))))
29	22	oveqd 7437	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
30		eqidd 2729	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → 𝑧 = 𝑧)
31	20, 29, 30	oveq123d 7441	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥 + 𝑦) · 𝑧))
32	20	oveqd 7437	. . . . . . . . . . . . 13 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (𝑦𝑡𝑧) = (𝑦 · 𝑧))
33	22, 26, 32	oveq123d 7441	. . . . . . . . . . . 12 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
34	31, 33	eqeq12d 2744	. . . . . . . . . . 11 ⊢ ((((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑝 = + ) ∧ 𝑡 = · ) → (((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧)) ↔ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))
35	28, 34	anbi12d 631	. . . . . . . . . 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)))))
36	19, 35	raleqbidv 3339	. . . . . . . . 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)))))
37	19, 36	raleqbidv 3339	. . . . . . . 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)))))
38	19, 37	raleqbidv 3339	. . . . . . 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)))))
39	14, 18, 38	sbcied2 3824	. . . . . 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)))))
40	9, 13, 39	sbcied2 3824	. . . . 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)))))
41	5, 8, 40	sbcied2 3824	. . . 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)))))
42	4, 41	anbi12d 631	. . 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))))))
43		df-ring 20175	. . 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))))}
44	42, 43	elrab2 3685	. 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))))))
45		3anass 1093	. 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))))))
46	44, 45	bitr4i 278	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:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3058  Vcvv 3471  [wsbc 3776  ‘cfv 6548  (class class class)co 7420  Basecbs 17180  +_gcplusg 17233  ._rcmulr 17234  Mndcmnd 18694  Grpcgrp 18890  mulGrpcmgp 20074  Ringcrg 20173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-ring 20175

This theorem is referenced by:  ringgrp  20178  ringmgp  20179  ringdilem  20189  ringrng  20221  isringrng  20223  ringpropd  20224  isringd  20227  ringsrg  20233  ring1  20246  prdsringd  20257  2zrngnring  47320  cznnring  47324