Theorem isdomn 19294

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		fvexd 6203	. . 3 |- ( r = R -> ( Base ` r ) e. _V )
2		fveq2 6191	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		isdomn.b	. . . 4 |- B = ( Base ` R )
4	2, 3	syl6eqr 2674	. . 3 |- ( r = R -> ( Base ` r ) = B )
5		fvexd 6203	. . . 4 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) e. _V )
6		fveq2 6191	. . . . . 6 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
7	6	adantr 481	. . . . 5 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) = ( 0g ` R ) )
8		isdomn.z	. . . . 5 |- .0. = ( 0g ` R )
9	7, 8	syl6eqr 2674	. . . 4 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) = .0. )
10		simplr 792	. . . . 5 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> b = B )
11		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
12		isdomn.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
13	11, 12	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> ( .r ` r ) = .x. )
14	13	oveqdr 6674	. . . . . . . 8 |- ( ( r = R /\ b = B ) -> ( x ( .r ` r ) y ) = ( x .x. y ) )
15		id 22	. . . . . . . 8 |- ( z = .0. -> z = .0. )
16	14, 15	eqeqan12d 2638	. . . . . . 7 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( ( x ( .r ` r ) y ) = z <-> ( x .x. y ) = .0. ) )
17		eqeq2 2633	. . . . . . . . 9 |- ( z = .0. -> ( x = z <-> x = .0. ) )
18		eqeq2 2633	. . . . . . . . 9 |- ( z = .0. -> ( y = z <-> y = .0. ) )
19	17, 18	orbi12d 746	. . . . . . . 8 |- ( z = .0. -> ( ( x = z \/ y = z ) <-> ( x = .0. \/ y = .0. ) ) )
20	19	adantl 482	. . . . . . 7 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( ( x = z \/ y = z ) <-> ( x = .0. \/ y = .0. ) ) )
21	16, 20	imbi12d 334	. . . . . 6 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( ( ( x ( .r ` r ) y ) = z -> ( x = z \/ y = z ) ) <-> ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
22	10, 21	raleqbidv 3152	. . . . 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. ) ) ) )
23	10, 22	raleqbidv 3152	. . . 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. ) ) ) )
24	5, 9, 23	sbcied2 3473	. . 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. ) ) ) )
25	1, 4, 24	sbcied2 3473	. 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. ) ) ) )
26		df-domn 19284	. 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 ) ) }
27	25, 26	elrab2 3366	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   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100  NzRingcnzr 19257  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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-iota 5851  df-fv 5896  df-ov 6653  df-domn 19284

This theorem is referenced by:  domnnzr  19295  domneq0  19297  isdomn2  19299  opprdomn  19301  abvn0b  19302  znfld  19909  ply1domn  23883  fta1b  23929  isdomn3  37782