Theorem islring 20568

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 6922	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2		islring.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3	1, 2	eqtr4di 2798	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
4		fveq2 6922	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
5		islring.a	. . . . . . . 8 ⊢ + = (+g‘𝑅)
6	4, 5	eqtr4di 2798	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + )
7	6	oveqd 7467	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥(+g‘𝑟)𝑦) = (𝑥 + 𝑦))
8		fveq2 6922	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
9		islring.1	. . . . . . 7 ⊢ 1 = (1r‘𝑅)
10	8, 9	eqtr4di 2798	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
11	7, 10	eqeq12d 2756	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) ↔ (𝑥 + 𝑦) = 1 ))
12		fveq2 6922	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
13		islring.u	. . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅)
14	12, 13	eqtr4di 2798	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
15	14	eleq2d 2830	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Unit‘𝑟) ↔ 𝑥 ∈ 𝑈))
16	14	eleq2d 2830	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑦 ∈ (Unit‘𝑟) ↔ 𝑦 ∈ 𝑈))
17	15, 16	orbi12d 917	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)) ↔ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
18	11, 17	imbi12d 344	. . . 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 3354	. . 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 3354	. 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 20567	. 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 3711	1 ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ‘cfv 6575  (class class class)co 7450  Basecbs 17260  +_gcplusg 17313  1_rcur 20210  Unitcui 20383  NzRingcnzr 20540  LRingclring 20566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-ov 7453  df-lring 20567

This theorem is referenced by:  lringuplu  20572