Theorem isdomn2OLD 20691

Description: Obsolete version of isdomn2 20690 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 2740	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		isdomn2.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	isdomn 20684	. 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
5		dfss3 3911	. . . 4 ⊢ ((𝐵 ∖ { 0 }) ⊆ 𝐸 ↔ ∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸)
6		isdomn2.t	. . . . . . . . 9 ⊢ 𝐸 = (RLReg‘𝑅)
7	6, 1, 2, 3	isrrg 20677	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐸 ↔ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
8	7	baib 540	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐸 ↔ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
9	8	imbi2d 341	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝑥 ≠ 0 → 𝑥 ∈ 𝐸) ↔ (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ))))
10	9	ralbiia 3084	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → 𝑥 ∈ 𝐸) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
11		eldifsn 4726	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∖ { 0 }) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ))
12	11	imbi1i 350	. . . . . . 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 276	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 ∖ { 0 }) → 𝑥 ∈ 𝐸) ↔ (𝑥 ∈ 𝐵 → (𝑥 ≠ 0 → 𝑥 ∈ 𝐸)))
15	14	ralbii2 3082	. . . . 5 ⊢ (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → 𝑥 ∈ 𝐸))
16		con34b 317	. . . . . . . . 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 991	. . . . . . . . . . 11 ⊢ (¬ (𝑥 = 0 ∨ 𝑦 = 0 ) ↔ (¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ))
19	18	imbi1i 350	. . . . . . . . . 10 ⊢ ((¬ (𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ ((¬ 𝑥 = 0 ∧ ¬ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ))
20		df-ne 2936	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0 )
21		con34b 317	. . . . . . . . . . 11 ⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 ) ↔ (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ))
22	20, 21	imbi12i 351	. . . . . . . . . 10 ⊢ ((𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )) ↔ (¬ 𝑥 = 0 → (¬ 𝑦 = 0 → ¬ (𝑥(.r‘𝑅)𝑦) = 0 )))
23	17, 19, 22	3bitr4i 304	. . . . . . . . 9 ⊢ ((¬ (𝑥 = 0 ∨ 𝑦 = 0 ) → ¬ (𝑥(.r‘𝑅)𝑦) = 0 ) ↔ (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
24	16, 23	bitri 276	. . . . . . . 8 ⊢ (((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝑥 ≠ 0 → ((𝑥(.r‘𝑅)𝑦) = 0 → 𝑦 = 0 )))
25	24	ralbii 3086	. . . . . . 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 3165	. . . . . . 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 276	. . . . . 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 3086	. . . . 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 304	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ { 0 })𝑥 ∈ 𝐸 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))
30	5, 29	bitr2i 277	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )) ↔ (𝐵 ∖ { 0 }) ⊆ 𝐸)
31	30	anbi2i 629	. 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 276	1 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054   ∖ cdif 3887   ⊆ wss 3890  {csn 4562  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  ._rcmulr 17219  0_gc0g 17400  NzRingcnzr 20491  RLRegcrlreg 20670  Domncdomn 20671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-rlreg 20673  df-domn 20674

This theorem is referenced by: (None)