Theorem domneq0 14410

Description: In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)

Hypotheses

Ref	Expression
domneq0.b	|- B = ( Base ` R )
domneq0.t	|- .x. = ( .r ` R )
domneq0.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
domneq0	|- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) )

Proof of Theorem domneq0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpc 1023	. . 3 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( X e. B /\ Y e. B ) )
2		domneq0.b	. . . . . 6 |- B = ( Base ` R )
3		domneq0.t	. . . . . 6 |- .x. = ( .r ` R )
4		domneq0.z	. . . . . 6 |- .0. = ( 0g ` R )
5	2, 3, 4	isdomn 14407	. . . . 5 |- ( R e. Domn <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) )
6	5	simprbi 275	. . . 4 |- ( R e. Domn -> A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) )
7	6	3ad2ant1 1045	. . 3 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) )
8		oveq1 6056	. . . . . 6 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
9	8	eqeq1d 2241	. . . . 5 |- ( x = X -> ( ( x .x. y ) = .0. <-> ( X .x. y ) = .0. ) )
10		eqeq1 2239	. . . . . 6 |- ( x = X -> ( x = .0. <-> X = .0. ) )
11	10	orbi1d 799	. . . . 5 |- ( x = X -> ( ( x = .0. \/ y = .0. ) <-> ( X = .0. \/ y = .0. ) ) )
12	9, 11	imbi12d 234	. . . 4 |- ( x = X -> ( ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) <-> ( ( X .x. y ) = .0. -> ( X = .0. \/ y = .0. ) ) ) )
13		oveq2 6057	. . . . . 6 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
14	13	eqeq1d 2241	. . . . 5 |- ( y = Y -> ( ( X .x. y ) = .0. <-> ( X .x. Y ) = .0. ) )
15		eqeq1 2239	. . . . . 6 |- ( y = Y -> ( y = .0. <-> Y = .0. ) )
16	15	orbi2d 798	. . . . 5 |- ( y = Y -> ( ( X = .0. \/ y = .0. ) <-> ( X = .0. \/ Y = .0. ) ) )
17	14, 16	imbi12d 234	. . . 4 |- ( y = Y -> ( ( ( X .x. y ) = .0. -> ( X = .0. \/ y = .0. ) ) <-> ( ( X .x. Y ) = .0. -> ( X = .0. \/ Y = .0. ) ) ) )
18	12, 17	rspc2va 2934	. . 3 |- ( ( ( X e. B /\ Y e. B ) /\ A. x e. B A. y e. B ( ( x .x. y ) = .0. -> ( x = .0. \/ y = .0. ) ) ) -> ( ( X .x. Y ) = .0. -> ( X = .0. \/ Y = .0. ) ) )
19	1, 7, 18	syl2anc 411	. 2 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( ( X .x. Y ) = .0. -> ( X = .0. \/ Y = .0. ) ) )
20		domnring 14409	. . . . . 6 |- ( R e. Domn -> R e. Ring )
21	20	3ad2ant1 1045	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> R e. Ring )
22		simp3 1026	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> Y e. B )
23	2, 3, 4	ringlz 14179	. . . . 5 |- ( ( R e. Ring /\ Y e. B ) -> ( .0. .x. Y ) = .0. )
24	21, 22, 23	syl2anc 411	. . . 4 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( .0. .x. Y ) = .0. )
25		oveq1 6056	. . . . 5 |- ( X = .0. -> ( X .x. Y ) = ( .0. .x. Y ) )
26	25	eqeq1d 2241	. . . 4 |- ( X = .0. -> ( ( X .x. Y ) = .0. <-> ( .0. .x. Y ) = .0. ) )
27	24, 26	syl5ibrcom 157	. . 3 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( X = .0. -> ( X .x. Y ) = .0. ) )
28		simp2 1025	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> X e. B )
29	2, 3, 4	ringrz 14180	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )
30	21, 28, 29	syl2anc 411	. . . 4 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( X .x. .0. ) = .0. )
31		oveq2 6057	. . . . 5 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
32	31	eqeq1d 2241	. . . 4 |- ( Y = .0. -> ( ( X .x. Y ) = .0. <-> ( X .x. .0. ) = .0. ) )
33	30, 32	syl5ibrcom 157	. . 3 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( Y = .0. -> ( X .x. Y ) = .0. ) )
34	27, 33	jaod 725	. 2 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( ( X = .0. \/ Y = .0. ) -> ( X .x. Y ) = .0. ) )
35	19, 34	impbid 129	1 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( ( X .x. Y ) = .0. <-> ( X = .0. \/ Y = .0. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   ` cfv 5351  (class class class)co 6049   Basecbs 13204   .rcmulr 13283   0gc0g 13461   Ringcrg 14132  NzRingcnzr 14316  Domncdomn 14393

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-plusg 13295  df-mulr 13296  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-minusg 13709  df-mgp 14057  df-ring 14134  df-nzr 14317  df-domn 14396

This theorem is referenced by:  domnmuln0  14411  znidomb  14798