Theorem isdomn2OLD 20729

Description: Obsolete version of isdomn2 20728 as of 21-Jun-2025. (Contributed by Mario Carneiro, 28-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
isdomn2OLD	⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))

Proof of Theorem isdomn2OLD

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 2735	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		isdomn2.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	isdomn 20722	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
5		dfss3 3984	. . . 4 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸)
6		isdomn2.t	. . . . . . . . 9 ⊢ 𝐸 = (RLReg‘𝑅)
7	6, 1, 2, 3	isrrg 20715	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐸 ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
8	7	baib 535	. . . . . . 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 3089	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → 𝑥 ∈ 𝐸) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
11		eldifsn 4791	. . . . . . . 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 450	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 𝑥 ∈ 𝐸) ↔ (𝑥 ∈ 𝐵 → (𝑥 ≠ 0 → 𝑥 ∈ 𝐸)))
14	12, 13	bitri 275	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐸) ↔ (𝑥 ∈ 𝐵 → (𝑥 ≠ 0 → 𝑥 ∈ 𝐸)))
15	14	ralbii2 3087	. . . . 5 ⊢ (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → 𝑥 ∈ 𝐸))
16		con34b 316	. . . . . . . . 9 ⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (¬ (𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ))
17		impexp 450	. . . . . . . . . 10 ⊢ (((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 )))
18		ioran 985	. . . . . . . . . . 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 2939	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0 )
21		con34b 316	. . . . . . . . . . 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 303	. . . . . . . . 9 ⊢ ((¬ (𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
24	16, 23	bitri 275	. . . . . . . 8 ⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
25	24	ralbii 3091	. . . . . . 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 3178	. . . . . . 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 275	. . . . . 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 3091	. . . . 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 303	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))
30	5, 29	bitr2i 276	. . 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 275	1 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059   ∖ cdif 3960   ⊆ wss 3963  {csn 4631  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  ._rcmulr 17299  0_gc0g 17486  NzRingcnzr 20529  RLRegcrlreg 20708  Domncdomn 20709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-rlreg 20711  df-domn 20712

This theorem is referenced by: (None)