Theorem isdomn 14218

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 13077	. . . . 5 |- Base Fn _V
2		vex 2802	. . . . 5 |- r e. _V
3		funfvex 5640	. . . . . 6 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
4	3	funfni 5419	. . . . 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 5623	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
8		isdomn.b	. . . 4 |- B = ( Base ` R )
9	7, 8	eqtr4di 2280	. . 3 |- ( r = R -> ( Base ` r ) = B )
10		fn0g 13394	. . . . . 6 |- 0g Fn _V
11		funfvex 5640	. . . . . . 7 |- ( ( Fun 0g /\ r e. dom 0g ) -> ( 0g ` r ) e. _V )
12	11	funfni 5419	. . . . . 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 5623	. . . . . 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 2280	. . . 4 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) = .0. )
19		simplr 528	. . . . 5 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> b = B )
20		fveq2 5623	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
21		isdomn.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
22	20, 21	eqtr4di 2280	. . . . . . . . 9 |- ( r = R -> ( .r ` r ) = .x. )
23	22	oveqdr 6022	. . . . . . . 8 |- ( ( r = R /\ b = B ) -> ( x ( .r ` r ) y ) = ( x .x. y ) )
24		id 19	. . . . . . . 8 |- ( z = .0. -> z = .0. )
25	23, 24	eqeqan12d 2245	. . . . . . 7 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( ( x ( .r ` r ) y ) = z <-> ( x .x. y ) = .0. ) )
26		eqeq2 2239	. . . . . . . . 9 |- ( z = .0. -> ( x = z <-> x = .0. ) )
27		eqeq2 2239	. . . . . . . . 9 |- ( z = .0. -> ( y = z <-> y = .0. ) )
28	26, 27	orbi12d 798	. . . . . . . 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 2744	. . . . 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 2744	. . . 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 3066	. . 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 3066	. 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 14208	. 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 2962	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 713   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   [.wsbc 3028   Fn wfn 5309   ` cfv 5314  (class class class)co 5994   Basecbs 13018   .rcmulr 13097   0gc0g 13275  NzRingcnzr 14128  Domncdomn 14205

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-riota 5947  df-ov 5997  df-inn 9099  df-ndx 13021  df-slot 13022  df-base 13024  df-0g 13277  df-domn 14208

This theorem is referenced by:  domnnzr  14219  domneq0  14221  opprdomnbg  14223  znidom  14606