Theorem isdomn 20873

Description: Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)

Hypotheses

Ref	Expression
isdomn.b	⊢ 𝐵 = (Base‘𝑅)
isdomn.t	⊢ · = (.r‘𝑅)
isdomn.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
isdomn	⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥, 0 ,𝑦

Allowed substitution hints:   · (𝑥, 𝑦)

Proof of Theorem isdomn

Dummy variables 𝑏 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6897	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
2		fveq2 6882	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		isdomn.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2815	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5		fvexd 6897	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (0g‘𝑟) ∈ V)
6		fveq2 6882	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
7	6	adantr 486	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (0g‘𝑟) = (0g‘𝑅))
8		isdomn.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
9	7, 8	eqtr4di 2815	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (0g‘𝑟) = 0 )
10		simplr 781	. . . . 5 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → 𝑏 = 𝐵)
11		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
12		isdomn.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
13	11, 12	eqtr4di 2815	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
14	13	oveqdr 7445	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
15		id 23	. . . . . . . 8 ⊢ (𝑧 = 0 → 𝑧 = 0 )
16	14, 15	eqeqan12d 2776	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥(.r‘𝑟)𝑦) = 𝑧 ↔ (𝑥 · 𝑦) = 0 ))
17		eqeq2 2774	. . . . . . . . 9 ⊢ (𝑧 = 0 → (𝑥 = 𝑧 ↔ 𝑥 = 0 ))
18		eqeq2 2774	. . . . . . . . 9 ⊢ (𝑧 = 0 → (𝑦 = 𝑧 ↔ 𝑦 = 0 ))
19	17, 18	orbi12d 932	. . . . . . . 8 ⊢ (𝑧 = 0 → ((𝑥 = 𝑧 ∨ 𝑦 = 𝑧) ↔ (𝑥 = 0 ∨ 𝑦 = 0 )))
20	19	adantl 487	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥 = 𝑧 ∨ 𝑦 = 𝑧) ↔ (𝑥 = 0 ∨ 𝑦 = 0 )))
21	16, 20	imbi12d 347	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧)) ↔ ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
22	10, 21	raleqbidv 3336	. . . . 5 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧)) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
23	10, 22	raleqbidv 3336	. . . 4 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
24	5, 9, 23	sbcied2 3786	. . 3 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → ([(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
25	1, 4, 24	sbcied2 3786	. 2 ⊢ (𝑟 = 𝑅 → ([(Base‘𝑟) / 𝑏][(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
26		df-domn 20863	. 2 ⊢ Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧))}
27	25, 26	elrab2 3652	1 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453  [wsbc 3742  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  ._rcmulr 17349  0_gc0g 17530  NzRingcnzr 20678  Domncdomn 20860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420  df-domn 20863

This theorem is used by:  domnnzr  20874  domneq0  20876  isdomn2  20879  isdomn3  20882  isdomn4  20883  opprdomnb  20884  abvn0b  21008  prmidl0  21547  qsidomlem2  21550  znfld  21779  ply1domn  26356  fta1b  26404  domnpropd  33728  subrdom  33733  ricdomn1  33737  mplidomlem  34045