Theorem isdomn 13765

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

Hypotheses

Ref	Expression
isdomn.b	|- B = ( Base ` R )
isdomn.t	|- .x. = ( .r ` R )
isdomn.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
isdomn	|- ( R e. Domn <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )

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

Allowed substitution hints:   .x. ( 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 12676	. . . . 5 |- Base Fn _V
2		vex 2763	. . . . 5 |- r e. _V
3		funfvex 5571	. . . . . 6 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
4	3	funfni 5354	. . . . 5 |- ( ( Base Fn _V /\ r e. _V ) -> ( Base ` r ) e. _V )
5	1, 2, 4	mp2an 426	. . . 4 |- ( Base ` r ) e. _V
6	5	a1i 9	. . 3 |- ( r = R -> ( Base ` r ) e. _V )
7		fveq2 5554	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
8		isdomn.b	. . . 4 |- B = ( Base ` R )
9	7, 8	eqtr4di 2244	. . 3 |- ( r = R -> ( Base ` r ) = B )
10		fn0g 12958	. . . . . 6 |- 0g Fn _V
11		funfvex 5571	. . . . . . 7 |- ( ( Fun 0g /\ r e. dom 0g ) -> ( 0g ` r ) e. _V )
12	11	funfni 5354	. . . . . 6 |- ( ( 0g Fn _V /\ r e. _V ) -> ( 0g ` r ) e. _V )
13	10, 2, 12	mp2an 426	. . . . 5 |- ( 0g ` r ) e. _V
14	13	a1i 9	. . . 4 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) e. _V )
15		fveq2 5554	. . . . . 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 2244	. . . 4 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) = .0. )
19		simplr 528	. . . . 5 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> b = B )
20		fveq2 5554	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
21		isdomn.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
22	20, 21	eqtr4di 2244	. . . . . . . . 9 |- ( r = R -> ( .r ` r ) = .x. )
23	22	oveqdr 5946	. . . . . . . 8 |- ( ( r = R /\ b = B ) -> ( x ( .r ` r ) y ) = ( x .x. y ) )
24		id 19	. . . . . . . 8 |- ( z = .0. -> z = .0. )
25	23, 24	eqeqan12d 2209	. . . . . . 7 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( ( x ( .r ` r ) y ) = z <-> ( x .x. y ) = .0. ) )
26		eqeq2 2203	. . . . . . . . 9 |- ( z = .0. -> ( x = z <-> x = .0. ) )
27		eqeq2 2203	. . . . . . . . 9 |- ( z = .0. -> ( y = z <-> y = .0. ) )
28	26, 27	orbi12d 794	. . . . . . . 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 .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
31	19, 30	raleqbidv 2706	. . . . 5 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( A. y e. b ( ( x ( .r ` r ) y ) = z -> ( x = z \/ y = z ) ) <-> A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
32	19, 31	raleqbidv 2706	. . . 4 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( A. x e. b A. y e. b ( ( x ( .r ` r ) y ) = z -> ( x = z \/ y = z ) ) <-> A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
33	14, 18, 32	sbcied2 3023	. . 3 |- ( ( r = R /\ b = B ) -> ( [. ( 0g ` r ) / z ]. A. x e. b A. y e. b ( ( x ( .r ` r ) y ) = z -> ( x = z \/ y = z ) ) <-> A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
34	6, 9, 33	sbcied2 3023	. 2 |- ( r = R -> ( [. ( Base ` r ) / b ]. [. ( 0g ` r ) / z ]. A. x e. b A. y e. b ( ( x ( .r ` r ) y ) = z -> ( x = z \/ y = z ) ) <-> A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
35		df-domn 13755	. 2 |- Domn = { r e. NzRing | [. ( Base ` r ) / b ]. [. ( 0g ` r ) / z ]. A. x e. b A. y e. b ( ( x ( .r ` r ) y ) = z -> ( x = z \/ y = z ) ) }
36	34, 35	elrab2 2919	1 |- ( R e. Domn <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   A.wral 2472   _Vcvv 2760   [.wsbc 2985   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   Basecbs 12618   .rcmulr 12696   0gc0g 12867  NzRingcnzr 13675  Domncdomn 13752

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869  df-domn 13755

This theorem is referenced by:  domnnzr  13766  domneq0  13768  opprdomnbg  13770  znidom  14145