Theorem isdomn2 20619

Description: A ring is a domain iff all nonzero elements are regular elements. (Contributed by Mario Carneiro, 28-Mar-2015.) (Proof shortened by SN, 21-Jun-2025.)

Hypotheses

Ref	Expression
isdomn2.b	⊢ 𝐵 = (Base‘𝑅)
isdomn2.t	⊢ 𝐸 = (RLReg‘𝑅)
isdomn2.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
isdomn2	⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))

Proof of Theorem isdomn2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isdomn2.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		eqid 2730	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		isdomn2.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	isdomn 20613	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
5		eldifi 4079	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐵)
6		isdomn2.t	. . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅)
7	6, 1, 2, 3	isrrg 20606	. . . . . . 7 ⊢ (𝑥 ∈ 𝐸 ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
8	7	baib 535	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐸 ↔ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
9	5, 8	syl 17	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) → (𝑥 ∈ 𝐸 ↔ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
10	9	ralbiia 3074	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))
11		dfss3 3921	. . . 4 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸)
12		isdomn5 20618	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))
13	10, 11, 12	3bitr4ri 304	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝐵 ∖ { 0 }) ⊆ 𝐸)
14	13	anbi2i 623	. 2 ⊢ ((𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
15	4, 14	bitri 275	1 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2110  ∀wral 3045   ∖ cdif 3897   ⊆ wss 3900  {csn 4574  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  ._rcmulr 17154  0_gc0g 17335  NzRingcnzr 20420  RLRegcrlreg 20599  Domncdomn 20600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-rlreg 20602  df-domn 20603

This theorem is referenced by:  domnrrg  20621  isdomn6  20622  drngdomn  20657  zringidom  33506