Theorem domneq0 13746

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 998	. . 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 13743	. . . . 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 1020	. . 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 5917	. . . . . 6 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
9	8	eqeq1d 2202	. . . . 5 |- ( x = X -> ( ( x .x. y ) = .0. <-> ( X .x. y ) = .0. ) )
10		eqeq1 2200	. . . . . 6 |- ( x = X -> ( x = .0. <-> X = .0. ) )
11	10	orbi1d 792	. . . . 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 5918	. . . . . 6 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
14	13	eqeq1d 2202	. . . . 5 |- ( y = Y -> ( ( X .x. y ) = .0. <-> ( X .x. Y ) = .0. ) )
15		eqeq1 2200	. . . . . 6 |- ( y = Y -> ( y = .0. <-> Y = .0. ) )
16	15	orbi2d 791	. . . . 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 2878	. . 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 13745	. . . . . 6 |- ( R e. Domn -> R e. Ring )
21	20	3ad2ant1 1020	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> R e. Ring )
22		simp3 1001	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> Y e. B )
23	2, 3, 4	ringlz 13517	. . . . 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 5917	. . . . 5 |- ( X = .0. -> ( X .x. Y ) = ( .0. .x. Y ) )
26	25	eqeq1d 2202	. . . 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 1000	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> X e. B )
29	2, 3, 4	ringrz 13518	. . . . 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 5918	. . . . 5 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
32	31	eqeq1d 2202	. . . 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 718	. 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 709   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   ` cfv 5246  (class class class)co 5910   Basecbs 12605   .rcmulr 12683   0gc0g 12854   Ringcrg 13470  NzRingcnzr 13653  Domncdomn 13730

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-pre-ltirr 7974  ax-pre-ltadd 7978

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-pnf 8046  df-mnf 8047  df-ltxr 8049  df-inn 8973  df-2 9031  df-3 9032  df-ndx 12608  df-slot 12609  df-base 12611  df-sets 12612  df-plusg 12695  df-mulr 12696  df-0g 12856  df-mgm 12926  df-sgrp 12972  df-mnd 12985  df-grp 13062  df-minusg 13063  df-mgp 13395  df-ring 13472  df-nzr 13654  df-domn 13733

This theorem is referenced by:  domnmuln0  13747  znidomb  14117