Theorem isrngod 33697

Description: Conditions that determine a ring. (Changed label from isringd 18585 to isrngod 33697-NM 2-Aug-2013.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isringod.1	⊢ (𝜑 → 𝐺 ∈ AbelOp)
isringod.2	⊢ (𝜑 → 𝑋 = ran 𝐺)
isringod.3	⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋)
isringod.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
isringod.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
isringod.6	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
isringod.7	⊢ (𝜑 → 𝑈 ∈ 𝑋)
isringod.8	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = 𝑦)
isringod.9	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦)

Assertion

Ref	Expression
isrngod	⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)

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

Allowed substitution hint:   𝑈(𝑧)

Proof of Theorem isrngod

Step	Hyp	Ref	Expression
1		isringod.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ AbelOp)
2		isringod.3	. . . 4 ⊢ (𝜑 → 𝐻:(𝑋 × 𝑋)⟶𝑋)
3		isringod.2	. . . . . 6 ⊢ (𝜑 → 𝑋 = ran 𝐺)
4	3	sqxpeqd 5141	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
5	4, 3	feq23d 6040	. . . 4 ⊢ (𝜑 → (𝐻:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺))
6	2, 5	mpbid 222	. . 3 ⊢ (𝜑 → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
7		isringod.4	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
8		isringod.5	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
9		isringod.6	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))
10	7, 8, 9	3jca 1242	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
11	10	ralrimivvva 2972	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
12	3	raleqdv 3144	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ ∀𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
13	3, 12	raleqbidv 3152	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ ∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
14	3, 13	raleqbidv 3152	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ↔ ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧)))))
15	11, 14	mpbid 222	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
16		isringod.7	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑋)
17		isringod.8	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑈𝐻𝑦) = 𝑦)
18		isringod.9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑦𝐻𝑈) = 𝑦)
19	17, 18	jca 554	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦))
20	19	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦))
21		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦))
22	21	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑥 = 𝑈 → ((𝑥𝐻𝑦) = 𝑦 ↔ (𝑈𝐻𝑦) = 𝑦))
23		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = 𝑈 → (𝑦𝐻𝑥) = (𝑦𝐻𝑈))
24	23	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑥 = 𝑈 → ((𝑦𝐻𝑥) = 𝑦 ↔ (𝑦𝐻𝑈) = 𝑦))
25	22, 24	anbi12d 747	. . . . . . . 8 ⊢ (𝑥 = 𝑈 → (((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦)))
26	25	ralbidv 2986	. . . . . . 7 ⊢ (𝑥 = 𝑈 → (∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝑋 ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦)))
27	26	rspcev 3309	. . . . . 6 ⊢ ((𝑈 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑋 ((𝑈𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑈) = 𝑦)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
28	16, 20, 27	syl2anc 693	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
29	3	raleqdv 3144	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
30	3, 29	rexeqbidv 3153	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
31	28, 30	mpbid 222	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
32	15, 31	jca 554	. . 3 ⊢ (𝜑 → (∀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)))
33	1, 6, 32	jca31 557	. 2 ⊢ (𝜑 → ((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))))
34		rnexg 7098	. . . . . 6 ⊢ (𝐺 ∈ AbelOp → ran 𝐺 ∈ V)
35	1, 34	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ V)
36		xpexg 6960	. . . . 5 ⊢ ((ran 𝐺 ∈ V ∧ ran 𝐺 ∈ V) → (ran 𝐺 × ran 𝐺) ∈ V)
37	35, 35, 36	syl2anc 693	. . . 4 ⊢ (𝜑 → (ran 𝐺 × ran 𝐺) ∈ V)
38		fex 6490	. . . 4 ⊢ ((𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ (ran 𝐺 × ran 𝐺) ∈ V) → 𝐻 ∈ V)
39	6, 37, 38	syl2anc 693	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
40		eqid 2622	. . . 4 ⊢ ran 𝐺 = ran 𝐺
41	40	isrngo 33696	. . 3 ⊢ (𝐻 ∈ V → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((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)))))
42	39, 41	syl 17	. 2 ⊢ (𝜑 → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((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)))))
43	33, 42	mpbird 247	1 ⊢ (𝜑 → ⟨𝐺, 𝐻⟩ ∈ RingOps)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200  ⟨cop 4183   × cxp 5112  ran crn 5115  ⟶wf 5884  (class class class)co 6650  AbelOpcablo 27398  RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-rngo 33694

This theorem is referenced by:  iscringd  33797