Theorem isdomn 14345

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		basfn 13202	. . . . 5 ⊢ Base Fn V
2		vex 2806	. . . . 5 ⊢ 𝑟 ∈ V
3		funfvex 5665	. . . . . 6 ⊢ ((Fun Base ∧ 𝑟 ∈ dom Base) → (Base‘𝑟) ∈ V)
4	3	funfni 5439	. . . . 5 ⊢ ((Base Fn V ∧ 𝑟 ∈ V) → (Base‘𝑟) ∈ V)
5	1, 2, 4	mp2an 426	. . . 4 ⊢ (Base‘𝑟) ∈ V
6	5	a1i 9	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
7		fveq2 5648	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
8		isdomn.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
9	7, 8	eqtr4di 2282	. . 3 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
10		fn0g 13519	. . . . . 6 ⊢ 0g Fn V
11		funfvex 5665	. . . . . . 7 ⊢ ((Fun 0g ∧ 𝑟 ∈ dom 0g) → (0g‘𝑟) ∈ V)
12	11	funfni 5439	. . . . . 6 ⊢ ((0g Fn V ∧ 𝑟 ∈ V) → (0g‘𝑟) ∈ V)
13	10, 2, 12	mp2an 426	. . . . 5 ⊢ (0g‘𝑟) ∈ V
14	13	a1i 9	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (0g‘𝑟) ∈ V)
15		fveq2 5648	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
16	15	adantr 276	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (0g‘𝑟) = (0g‘𝑅))
17		isdomn.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
18	16, 17	eqtr4di 2282	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (0g‘𝑟) = 0 )
19		simplr 529	. . . . 5 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → 𝑏 = 𝐵)
20		fveq2 5648	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
21		isdomn.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
22	20, 21	eqtr4di 2282	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
23	22	oveqdr 6056	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
24		id 19	. . . . . . . 8 ⊢ (𝑧 = 0 → 𝑧 = 0 )
25	23, 24	eqeqan12d 2247	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥(.r‘𝑟)𝑦) = 𝑧 ↔ (𝑥 · 𝑦) = 0 ))
26		eqeq2 2241	. . . . . . . . 9 ⊢ (𝑧 = 0 → (𝑥 = 𝑧 ↔ 𝑥 = 0 ))
27		eqeq2 2241	. . . . . . . . 9 ⊢ (𝑧 = 0 → (𝑦 = 𝑧 ↔ 𝑦 = 0 ))
28	26, 27	orbi12d 801	. . . . . . . 8 ⊢ (𝑧 = 0 → ((𝑥 = 𝑧 ∨ 𝑦 = 𝑧) ↔ (𝑥 = 0 ∨ 𝑦 = 0 )))
29	28	adantl 277	. . . . . . 7 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → ((𝑥 = 𝑧 ∨ 𝑦 = 𝑧) ↔ (𝑥 = 0 ∨ 𝑦 = 0 )))
30	25, 29	imbi12d 234	. . . . . 6 ⊢ (((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) ∧ 𝑧 = 0 ) → (((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧)) ↔ ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
31	19, 30	raleqbidv 2747	. . . . 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 ))))
32	19, 31	raleqbidv 2747	. . . 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 ))))
33	14, 18, 32	sbcied2 3070	. . 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 ))))
34	6, 9, 33	sbcied2 3070	. 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 ))))
35		df-domn 14335	. 2 ⊢ Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧))}
36	34, 35	elrab2 2966	1 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803  [wsbc 3032   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028  Basecbs 13143  ._rcmulr 13222  0_gc0g 13400  NzRingcnzr 14255  Domncdomn 14332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9187  df-ndx 13146  df-slot 13147  df-base 13149  df-0g 13402  df-domn 14335

This theorem is referenced by:  domnnzr  14346  domneq0  14348  opprdomnbg  14350  znidom  14733