Theorem islring 13688

Description: The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)

Hypotheses

Ref	Expression
islring.b	⊢ 𝐵 = (Base‘𝑅)
islring.a	⊢ + = (+g‘𝑅)
islring.1	⊢ 1 = (1r‘𝑅)
islring.u	⊢ 𝑈 = (Unit‘𝑅)

Assertion

Ref	Expression
islring	⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   + (𝑥,𝑦)   𝑈(𝑥,𝑦)   1 (𝑥,𝑦)

Proof of Theorem islring

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5554	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		islring.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	1, 2	eqtr4di 2244	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
4		fveq2 5554	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
5		islring.a	. . . . . . . 8 ⊢ + = (+g‘𝑅)
6	4, 5	eqtr4di 2244	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + )
7	6	oveqd 5935	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥(+g‘𝑟)𝑦) = (𝑥 + 𝑦))
8		fveq2 5554	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
9		islring.1	. . . . . . 7 ⊢ 1 = (1r‘𝑅)
10	8, 9	eqtr4di 2244	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
11	7, 10	eqeq12d 2208	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) ↔ (𝑥 + 𝑦) = 1 ))
12		fveq2 5554	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
13		islring.u	. . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅)
14	12, 13	eqtr4di 2244	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
15	14	eleq2d 2263	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Unit‘𝑟) ↔ 𝑥 ∈ 𝑈))
16	14	eleq2d 2263	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑦 ∈ (Unit‘𝑟) ↔ 𝑦 ∈ 𝑈))
17	15, 16	orbi12d 794	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)) ↔ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
18	11, 17	imbi12d 234	. . . 4 ⊢ (𝑟 = 𝑅 → (((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟))) ↔ ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
19	3, 18	raleqbidv 2706	. . 3 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟))) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
20	3, 19	raleqbidv 2706	. 2 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
21		df-lring 13687	. 2 ⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
22	20, 21	elrab2 2919	1 ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  +_gcplusg 12695  1_rcur 13455  Unitcui 13583  NzRingcnzr 13675  LRingclring 13686

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921  df-lring 13687

This theorem is referenced by:  lringuplu  13692