Theorem isdomn2 19299

Description: A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)

Hypotheses

Ref	Expression
isdomn2.b	|- B = ( Base ` R )
isdomn2.t	|- E = (RLReg ` R )
isdomn2.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
isdomn2	|- ( R e. Domn <-> ( R e. NzRing /\ ( B \ { .0. } ) C_ E ) )

Proof of Theorem isdomn2

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 2622	. . 3 |- ( .r ` R ) = ( .r ` R )
3		isdomn2.z	. . 3 |- .0. = ( 0g ` R )
4	1, 2, 3	isdomn 19294	. 2 |- ( R e. Domn <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
5		dfss3 3592	. . . 4 |- ( ( B \ { .0. } ) C_ E <-> A. x e. ( B \ { .0. } ) x e. E )
6		isdomn2.t	. . . . . . . . 9 |- E = (RLReg ` R )
7	6, 1, 2, 3	isrrg 19288	. . . . . . . 8 |- ( x e. E <-> ( x e. B /\ A. y e. B ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
8	7	baib 944	. . . . . . 7 |- ( x e. B -> ( x e. E <-> A. y e. B ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
9	8	imbi2d 330	. . . . . 6 |- ( x e. B -> ( ( x =/= .0. -> x e. E ) <-> ( x =/= .0. -> A. y e. B ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) ) )
10	9	ralbiia 2979	. . . . 5 |- ( A. x e. B ( x =/= .0. -> x e. E ) <-> A. x e. B ( x =/= .0. -> A. y e. B ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
11		eldifsn 4317	. . . . . . . 8 |- ( x e. ( B \ { .0. } ) <-> ( x e. B /\ x =/= .0. ) )
12	11	imbi1i 339	. . . . . . 7 |- ( ( x e. ( B \ { .0. } ) -> x e. E ) <-> ( ( x e. B /\ x =/= .0. ) -> x e. E ) )
13		impexp 462	. . . . . . 7 |- ( ( ( x e. B /\ x =/= .0. ) -> x e. E ) <-> ( x e. B -> ( x =/= .0. -> x e. E ) ) )
14	12, 13	bitri 264	. . . . . 6 |- ( ( x e. ( B \ { .0. } ) -> x e. E ) <-> ( x e. B -> ( x =/= .0. -> x e. E ) ) )
15	14	ralbii2 2978	. . . . 5 |- ( A. x e. ( B \ { .0. } ) x e. E <-> A. x e. B ( x =/= .0. -> x e. E ) )
16		con34b 306	. . . . . . . . 9 |- ( ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) <-> ( -. ( x = .0. \/ y = .0. ) -> -. ( x ( .r ` R ) y ) = .0. ) )
17		impexp 462	. . . . . . . . . 10 |- ( ( ( -. x = .0. /\ -. y = .0. ) -> -. ( x ( .r ` R ) y ) = .0. ) <-> ( -. x = .0. -> ( -. y = .0. -> -. ( x ( .r ` R ) y ) = .0. ) ) )
18		ioran 511	. . . . . . . . . . 11 |- ( -. ( x = .0. \/ y = .0. ) <-> ( -. x = .0. /\ -. y = .0. ) )
19	18	imbi1i 339	. . . . . . . . . 10 |- ( ( -. ( x = .0. \/ y = .0. ) -> -. ( x ( .r ` R ) y ) = .0. ) <-> ( ( -. x = .0. /\ -. y = .0. ) -> -. ( x ( .r ` R ) y ) = .0. ) )
20		df-ne 2795	. . . . . . . . . . 11 |- ( x =/= .0. <-> -. x = .0. )
21		con34b 306	. . . . . . . . . . 11 |- ( ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) <-> ( -. y = .0. -> -. ( x ( .r ` R ) y ) = .0. ) )
22	20, 21	imbi12i 340	. . . . . . . . . 10 |- ( ( x =/= .0. -> ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) <-> ( -. x = .0. -> ( -. y = .0. -> -. ( x ( .r ` R ) y ) = .0. ) ) )
23	17, 19, 22	3bitr4i 292	. . . . . . . . 9 |- ( ( -. ( x = .0. \/ y = .0. ) -> -. ( x ( .r ` R ) y ) = .0. ) <-> ( x =/= .0. -> ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
24	16, 23	bitri 264	. . . . . . . 8 |- ( ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) <-> ( x =/= .0. -> ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
25	24	ralbii 2980	. . . . . . 7 |- ( A. y e. B ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) <-> A. y e. B ( x =/= .0. -> ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
26		r19.21v 2960	. . . . . . 7 |- ( A. y e. B ( x =/= .0. -> ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) <-> ( x =/= .0. -> A. y e. B ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
27	25, 26	bitri 264	. . . . . 6 |- ( A. y e. B ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) <-> ( x =/= .0. -> A. y e. B ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
28	27	ralbii 2980	. . . . 5 |- ( A. x e. B A. y e. B ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) <-> A. x e. B ( x =/= .0. -> A. y e. B ( ( x ( .r ` R ) y ) = .0. -> y = .0. ) ) )
29	10, 15, 28	3bitr4i 292	. . . 4 |- ( A. x e. ( B \ { .0. } ) x e. E <-> A. x e. B A. y e. B ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) )
30	5, 29	bitr2i 265	. . 3 |- ( A. x e. B A. y e. B ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) <-> ( B \ { .0. } ) C_ E )
31	30	anbi2i 730	. 2 |- ( ( R e. NzRing /\ A. x e. B A. y e. B ( ( x ( .r ` R ) y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) <-> ( R e. NzRing /\ ( B \ { .0. } ) C_ E ) )
32	4, 31	bitri 264	1 |- ( R e. Domn <-> ( R e. NzRing /\ ( B \ { .0. } ) C_ E ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100  NzRingcnzr 19257  RLRegcrlreg 19279  Domncdomn 19280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-rlreg 19283  df-domn 19284

This theorem is referenced by:  domnrrg  19300  drngdomn  19303