Theorem isrngo 35792

Description: The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isring.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
isrngo	⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝐻,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

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

Proof of Theorem isrngo

Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5054	. . . 4 ⊢ (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps)
2		relrngo 35791	. . . . 5 ⊢ Rel RingOps
3	2	brrelex1i 5605	. . . 4 ⊢ (𝐺RingOps𝐻 → 𝐺 ∈ V)
4	1, 3	sylbir 238	. . 3 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V)
5	4	a1i 11	. 2 ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ V))
6		elex 3426	. . . 4 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V)
7	6	ad2antrr 726	. . 3 ⊢ (((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) → 𝐺 ∈ V)
8	7	a1i 11	. 2 ⊢ (𝐻 ∈ 𝐴 → (((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))) → 𝐺 ∈ V))
9		df-rngo 35790	. . . . 5 ⊢ RingOps = {⟨𝑔, ℎ⟩ ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))}
10	9	eleq2i 2829	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))})
11		simpl 486	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → 𝑔 = 𝐺)
12	11	eleq1d 2822	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑔 ∈ AbelOp ↔ 𝐺 ∈ AbelOp))
13		simpr 488	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ℎ = 𝐻)
14	11	rneqd 5807	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ran 𝑔 = ran 𝐺)
15		isring.1	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
16	14, 15	eqtr4di 2796	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ran 𝑔 = 𝑋)
17	16	sqxpeqd 5583	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (ran 𝑔 × ran 𝑔) = (𝑋 × 𝑋))
18	13, 17, 16	feq123d 6534	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔 ↔ 𝐻:(𝑋 × 𝑋)⟶𝑋))
19	12, 18	anbi12d 634	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ↔ (𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋)))
20	13	oveqd 7230	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
21		eqidd 2738	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → 𝑧 = 𝑧)
22	13, 20, 21	oveq123d 7234	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ𝑦)ℎ𝑧) = ((𝑥𝐻𝑦)𝐻𝑧))
23		eqidd 2738	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → 𝑥 = 𝑥)
24	13	oveqd 7230	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑦ℎ𝑧) = (𝑦𝐻𝑧))
25	13, 23, 24	oveq123d 7234	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥ℎ(𝑦ℎ𝑧)) = (𝑥𝐻(𝑦𝐻𝑧)))
26	22, 25	eqeq12d 2753	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ↔ ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
27	11	oveqd 7230	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
28	13, 23, 27	oveq123d 7234	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥ℎ(𝑦𝑔𝑧)) = (𝑥𝐻(𝑦𝐺𝑧)))
29	13	oveqd 7230	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥ℎ𝑧) = (𝑥𝐻𝑧))
30	11, 20, 29	oveq123d 7234	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
31	28, 30	eqeq12d 2753	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ↔ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧))))
32	11	oveqd 7230	. . . . . . . . . . . . 13 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
33	13, 32, 21	oveq123d 7234	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥𝐺𝑦)𝐻𝑧))
34	11, 29, 24	oveq123d 7234	. . . . . . . . . . . 12 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧)) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
35	33, 34	eqeq12d 2753	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧)) ↔ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
36	26, 31, 35	3anbi123d 1438	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ↔ (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
37	16, 36	raleqbidv 3313	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ↔ ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
38	16, 37	raleqbidv 3313	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
39	16, 38	raleqbidv 3313	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
40	20	eqeq1d 2739	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑥ℎ𝑦) = 𝑦 ↔ (𝑥𝐻𝑦) = 𝑦))
41	13	oveqd 7230	. . . . . . . . . . 11 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑦ℎ𝑥) = (𝑦𝐻𝑥))
42	41	eqeq1d 2739	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑦ℎ𝑥) = 𝑦 ↔ (𝑦𝐻𝑥) = 𝑦))
43	40, 42	anbi12d 634	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦) ↔ ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
44	16, 43	raleqbidv 3313	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
45	16, 44	rexeqbidv 3314	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦) ↔ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
46	39, 45	anbi12d 634	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)) ↔ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
47	19, 46	anbi12d 634	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦))) ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
48	47	opelopabga 5414	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑔, ℎ⟩ ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))} ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
49	10, 48	syl5bb 286	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ 𝐴) → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
50	49	expcom 417	. 2 ⊢ (𝐻 ∈ 𝐴 → (𝐺 ∈ V → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))))
51	5, 8, 50	pm5.21ndd 384	1 ⊢ (𝐻 ∈ 𝐴 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  Vcvv 3408  ⟨cop 4547   class class class wbr 5053  {copab 5115   × cxp 5549  ran crn 5552  ⟶wf 6376  (class class class)co 7213  AbelOpcablo 28625  RingOpscrngo 35789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-rngo 35790

This theorem is referenced by:  isrngod  35793  rngoi  35794