Theorem isdomn2 20668

Description: A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)

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 20663	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
5		dfss3 3919	. . . 4 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸)
6		isdomn2.t	. . . . . . . . 9 ⊢ 𝐸 = (RLReg‘𝑅)
7	6, 1, 2, 3	isrrg 20657	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐸 ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
8	7	baib 536	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐸 ↔ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
9	8	imbi2d 340	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝑥 ≠ 0 → 𝑥 ∈ 𝐸) ↔ (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))))
10	9	ralbiia 3090	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → 𝑥 ∈ 𝐸) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
11		eldifsn 4733	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ))
12	11	imbi1i 349	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐸) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝐸))
13		impexp 451	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝐸) ↔ (𝑥 ∈ 𝐵 → (𝑥 ≠ 0 → 𝑥 ∈ 𝐸)))
14	12, 13	bitri 274	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐸) ↔ (𝑥 ∈ 𝐵 → (𝑥 ≠ 0 → 𝑥 ∈ 𝐸)))
15	14	ralbii2 3088	. . . . 5 ⊢ (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → 𝑥 ∈ 𝐸))
16		con34b 315	. . . . . . . . 9 ⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (¬ (𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ))
17		impexp 451	. . . . . . . . . 10 ⊢ (((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 )))
18		ioran 981	. . . . . . . . . . 11 ⊢ (¬ (𝑥 = 0 ∨ 𝑦 = 0 ) ↔ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ))
19	18	imbi1i 349	. . . . . . . . . 10 ⊢ ((¬ (𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ ((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ))
20		df-ne 2941	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0 )
21		con34b 315	. . . . . . . . . . 11 ⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ) ↔ (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ))
22	20, 21	imbi12i 350	. . . . . . . . . 10 ⊢ ((𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 )))
23	17, 19, 22	3bitr4i 302	. . . . . . . . 9 ⊢ ((¬ (𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
24	16, 23	bitri 274	. . . . . . . 8 ⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
25	24	ralbii 3092	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
26		r19.21v 3172	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )) ↔ (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
27	25, 26	bitri 274	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
28	27	ralbii 3092	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
29	10, 15, 28	3bitr4i 302	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))
30	5, 29	bitr2i 275	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝐵 ∖ { 0 }) ⊆ 𝐸)
31	30	anbi2i 623	. 2 ⊢ ((𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))) ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
32	4, 31	bitri 274	1 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1540   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061   ∖ cdif 3894   ⊆ wss 3897  {csn 4572  ‘cfv 6473  (class class class)co 7329  Basecbs 17001  ._rcmulr 17052  0_gc0g 17239  NzRingcnzr 20626  RLRegcrlreg 20648  Domncdomn 20649

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6425  df-fun 6475  df-fv 6481  df-ov 7332  df-rlreg 20652  df-domn 20653

This theorem is referenced by:  domnrrg  20669  drngdomn  20672