Theorem isdomn2 20688

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 2736	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		isdomn2.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	isdomn 20682	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
5		eldifi 4071	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐵)
6		isdomn2.t	. . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅)
7	6, 1, 2, 3	isrrg 20675	. . . . . . 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 3081	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))
11		dfss3 3910	. . . 4 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸)
12		isdomn5 20687	. . . 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 624	. 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 848   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ∖ cdif 3886   ⊆ wss 3889  {csn 4567  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  ._rcmulr 17221  0_gc0g 17402  NzRingcnzr 20489  RLRegcrlreg 20668  Domncdomn 20669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-rlreg 20671  df-domn 20672

This theorem is referenced by:  domnrrg  20690  isdomn6  20691  drngdomn  20726  zringidom  33611