Theorem islring 20708

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 group:   𝑥,𝑅,𝑦

Allowed substitution hints:   𝐵(𝑥, 𝑦)   + (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 6882	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		islring.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	1, 2	eqtr4di 2815	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
4		fveq2 6882	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
5		islring.a	. . . . . . . 8 ⊢ + = (+g‘𝑅)
6	4, 5	eqtr4di 2815	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + )
7	6	oveqd 7434	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥(+g‘𝑟)𝑦) = (𝑥 + 𝑦))
8		fveq2 6882	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
9		islring.1	. . . . . . 7 ⊢ 1 = (1r‘𝑅)
10	8, 9	eqtr4di 2815	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
11	7, 10	eqeq12d 2778	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) ↔ (𝑥 + 𝑦) = 1 ))
12		fveq2 6882	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
13		islring.u	. . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅)
14	12, 13	eqtr4di 2815	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
15	14	eleq2d 2848	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Unit‘𝑟) ↔ 𝑥 ∈ 𝑈))
16	14	eleq2d 2848	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑦 ∈ (Unit‘𝑟) ↔ 𝑦 ∈ 𝑈))
17	15, 16	orbi12d 932	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)) ↔ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
18	11, 17	imbi12d 347	. . . 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 3336	. . 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 3336	. 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 20707	. 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 3652	1 ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  +_gcplusg 17348  1_rcur 20326  Unitcui 20502  NzRingcnzr 20678  LRingclring 20706

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420  df-lring 20707

This theorem is used by:  lringuplu  20712  drnglring  33910  dflring2  33911