Theorem isdomn 14561

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 13394	. . . . 5 |- Base Fn _V
2		vex 2824	. . . . 5 |- r e. _V
3		funfvex 5710	. . . . . 6 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
4	3	funfni 5481	. . . . 5 |- ( ( Base Fn _V /\ r e. _V ) -> ( Base ` r ) e. _V )
5	1, 2, 4	mp2an 430	. . . 4 |- ( Base ` r ) e. _V
6	5	a1i 9	. . 3 |- ( r = R -> ( Base ` r ) e. _V )
7		fveq2 5693	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
8		isdomn.b	. . . 4 |- B = ( Base ` R )
9	7, 8	eqtr4di 2289	. . 3 |- ( r = R -> ( Base ` r ) = B )
10		fn0g 13678	. . . . . 6 |- 0g Fn _V
11		funfvex 5710	. . . . . . 7 |- ( ( Fun 0g /\ r e. dom 0g ) -> ( 0g ` r ) e. _V )
12	11	funfni 5481	. . . . . 6 |- ( ( 0g Fn _V /\ r e. _V ) -> ( 0g ` r ) e. _V )
13	10, 2, 12	mp2an 430	. . . . 5 |- ( 0g ` r ) e. _V
14	13	a1i 9	. . . 4 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) e. _V )
15		fveq2 5693	. . . . . 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 2289	. . . 4 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) = .0. )
19		simplr 533	. . . . 5 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> b = B )
20		fveq2 5693	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
21		isdomn.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
22	20, 21	eqtr4di 2289	. . . . . . . . 9 |- ( r = R -> ( .r ` r ) = .x. )
23	22	oveqdr 6107	. . . . . . . 8 |- ( ( r = R /\ b = B ) -> ( x ( .r ` r ) y ) = ( x .x. y ) )
24		id 19	. . . . . . . 8 |- ( z = .0. -> z = .0. )
25	23, 24	eqeqan12d 2254	. . . . . . 7 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( ( x ( .r ` r ) y ) = z <-> ( x .x. y ) = .0. ) )
26		eqeq2 2248	. . . . . . . . 9 |- ( z = .0. -> ( x = z <-> x = .0. ) )
27		eqeq2 2248	. . . . . . . . 9 |- ( z = .0. -> ( y = z <-> y = .0. ) )
28	26, 27	orbi12d 805	. . . . . . . 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 2765	. . . . 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 2765	. . . 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 3089	. . 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 3089	. 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 14550	. 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 2985	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 720   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   [.wsbc 3051   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   Basecbs 13335   .rcmulr 13415   0gc0g 13593  NzRingcnzr 14469  Domncdomn 14547

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-riota 6032  df-ov 6082  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-0g 13595  df-domn 14550

This theorem is referenced by:  domnnzr  14562  domneq0  14564  opprdomnbg  14566  znidom  14975