Theorem domneq0 13804

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 13801	. . . . 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 5929	. . . . . 6 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
9	8	eqeq1d 2205	. . . . 5 |- ( x = X -> ( ( x .x. y ) = .0. <-> ( X .x. y ) = .0. ) )
10		eqeq1 2203	. . . . . 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 5930	. . . . . 6 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
14	13	eqeq1d 2205	. . . . 5 |- ( y = Y -> ( ( X .x. y ) = .0. <-> ( X .x. Y ) = .0. ) )
15		eqeq1 2203	. . . . . 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 2882	. . 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 13803	. . . . . 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 13575	. . . . 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 5929	. . . . 5 |- ( X = .0. -> ( X .x. Y ) = ( .0. .x. Y ) )
26	25	eqeq1d 2205	. . . 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 13576	. . . . 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 5930	. . . . 5 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
32	31	eqeq1d 2205	. . . 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 2167   A.wral 2475   ` cfv 5258  (class class class)co 5922   Basecbs 12654   .rcmulr 12732   0gc0g 12903   Ringcrg 13528  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-in1 615  ax-in2 616  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-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7968  ax-resscn 7969  ax-1cn 7970  ax-1re 7971  ax-icn 7972  ax-addcl 7973  ax-addrcl 7974  ax-mulcl 7975  ax-addcom 7977  ax-addass 7979  ax-i2m1 7982  ax-0lt1 7983  ax-0id 7985  ax-rnegex 7986  ax-pre-ltirr 7989  ax-pre-ltadd 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  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-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  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-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8061  df-mnf 8062  df-ltxr 8064  df-inn 8988  df-2 9046  df-3 9047  df-ndx 12657  df-slot 12658  df-base 12660  df-sets 12661  df-plusg 12744  df-mulr 12745  df-0g 12905  df-mgm 12975  df-sgrp 13021  df-mnd 13034  df-grp 13111  df-minusg 13112  df-mgp 13453  df-ring 13530  df-nzr 13712  df-domn 13791

This theorem is referenced by:  domnmuln0  13805  znidomb  14190