Theorem domneq0 14583

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 1027	. . 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 14580	. . . . 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 1049	. . 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 6092	. . . . . 6 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
9	8	eqeq1d 2247	. . . . 5 |- ( x = X -> ( ( x .x. y ) = .0. <-> ( X .x. y ) = .0. ) )
10		eqeq1 2245	. . . . . 6 |- ( x = X -> ( x = .0. <-> X = .0. ) )
11	10	orbi1d 803	. . . . 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 6093	. . . . . 6 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
14	13	eqeq1d 2247	. . . . 5 |- ( y = Y -> ( ( X .x. y ) = .0. <-> ( X .x. Y ) = .0. ) )
15		eqeq1 2245	. . . . . 6 |- ( y = Y -> ( y = .0. <-> Y = .0. ) )
16	15	orbi2d 802	. . . . 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 2944	. . 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 415	. 2 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( ( X .x. Y ) = .0. -> ( X = .0. \/ Y = .0. ) ) )
20		domnring 14582	. . . . . 6 |- ( R e. Domn -> R e. Ring )
21	20	3ad2ant1 1049	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> R e. Ring )
22		simp3 1030	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> Y e. B )
23	2, 3, 4	ringlz 14350	. . . . 5 |- ( ( R e. Ring /\ Y e. B ) -> ( .0. .x. Y ) = .0. )
24	21, 22, 23	syl2anc 415	. . . 4 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( .0. .x. Y ) = .0. )
25		oveq1 6092	. . . . 5 |- ( X = .0. -> ( X .x. Y ) = ( .0. .x. Y ) )
26	25	eqeq1d 2247	. . . 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 1029	. . . . 5 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> X e. B )
29	2, 3, 4	ringrz 14351	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )
30	21, 28, 29	syl2anc 415	. . . 4 |- ( ( R e. Domn /\ X e. B /\ Y e. B ) -> ( X .x. .0. ) = .0. )
31		oveq2 6093	. . . . 5 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
32	31	eqeq1d 2247	. . . 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 729	. 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. ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   Basecbs 13354   .rcmulr 13434   0gc0g 13612   Ringcrg 14302  NzRingcnzr 14488  Domncdomn 14566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-plusg 13446  df-mulr 13447  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-mgp 14220  df-ring 14304  df-nzr 14489  df-domn 14569

This theorem is used by:  domnmuln0  14584  znidomb  14995