Theorem isdomn 13801

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 12712	. . . . 5 |- Base Fn _V
2		vex 2766	. . . . 5 |- r e. _V
3		funfvex 5575	. . . . . 6 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
4	3	funfni 5358	. . . . 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 5558	. . . 4 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
8		isdomn.b	. . . 4 |- B = ( Base ` R )
9	7, 8	eqtr4di 2247	. . 3 |- ( r = R -> ( Base ` r ) = B )
10		fn0g 12994	. . . . . 6 |- 0g Fn _V
11		funfvex 5575	. . . . . . 7 |- ( ( Fun 0g /\ r e. dom 0g ) -> ( 0g ` r ) e. _V )
12	11	funfni 5358	. . . . . 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 5558	. . . . . 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 2247	. . . 4 |- ( ( r = R /\ b = B ) -> ( 0g ` r ) = .0. )
19		simplr 528	. . . . 5 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> b = B )
20		fveq2 5558	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
21		isdomn.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
22	20, 21	eqtr4di 2247	. . . . . . . . 9 |- ( r = R -> ( .r ` r ) = .x. )
23	22	oveqdr 5950	. . . . . . . 8 |- ( ( r = R /\ b = B ) -> ( x ( .r ` r ) y ) = ( x .x. y ) )
24		id 19	. . . . . . . 8 |- ( z = .0. -> z = .0. )
25	23, 24	eqeqan12d 2212	. . . . . . 7 |- ( ( ( r = R /\ b = B ) /\ z = .0. ) -> ( ( x ( .r ` r ) y ) = z <-> ( x .x. y ) = .0. ) )
26		eqeq2 2206	. . . . . . . . 9 |- ( z = .0. -> ( x = z <-> x = .0. ) )
27		eqeq2 2206	. . . . . . . . 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 2709	. . . . 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 2709	. . . 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 3027	. . 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 3027	. 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 13791	. 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 2923	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 2167   A.wral 2475   _Vcvv 2763   [.wsbc 2989   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12654   .rcmulr 12732   0gc0g 12903  NzRingcnzr 13711  Domncdomn 13788

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7968  ax-resscn 7969  ax-1re 7971  ax-addrcl 7974

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8988  df-ndx 12657  df-slot 12658  df-base 12660  df-0g 12905  df-domn 13791

This theorem is referenced by:  domnnzr  13802  domneq0  13804  opprdomnbg  13806  znidom  14189