Theorem isdomn 20990

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 6926	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
2		fveq2 6911	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		isdomn.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2816	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5		fvexd 6926	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (0g‘𝑟) ∈ V)
6		fveq2 6911	. . . . . 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 2816	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (0g‘𝑟) = 0 )
10		simplr 781	. . . . 5 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → 𝑏 = 𝐵)
11		fveq2 6911	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
12		isdomn.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
13	11, 12	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
14	13	oveqdr 7474	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
15		id 23	. . . . . . . 8 ⊢ (𝑧 = 0 → 𝑧 = 0 )
16	14, 15	eqeqan12d 2777	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥(.r‘𝑟)𝑦) = 𝑧 ↔ (𝑥 · 𝑦) = 0 ))
17		eqeq2 2775	. . . . . . . . 9 ⊢ (𝑧 = 0 → (𝑥 = 𝑧 ↔ 𝑥 = 0 ))
18		eqeq2 2775	. . . . . . . . 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 3338	. . . . 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 3338	. . . 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 20980	. 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 2147  ∀wral 3080  Vcvv 3454  [wsbc 3742  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  ._rcmulr 17461  0_gc0g 17642  NzRingcnzr 20795  Domncdomn 20977

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-domn 20980

This theorem is used by:  domnnzr  20991  domneq0  20993  isdomn2  20996  isdomn3  20999  isdomn4  21000  opprdomnb  21001  abvn0b  21126  prmidl0  21667  qsidomlem2  21670  znfld  21899  ply1domn  26475  fta1b  26523  domnpropd  33917  subrdom  33922  ricdomn1  33926  mplidomlem  34242