Theorem ringinvnzdiv 14062

Description: In a unital ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)

Hypotheses

Ref	Expression
ringinvnzdiv.b	|- B = ( Base ` R )
ringinvnzdiv.t	|- .x. = ( .r ` R )
ringinvnzdiv.u	|- .1. = ( 1r ` R )
ringinvnzdiv.z	|- .0. = ( 0g ` R )
ringinvnzdiv.r	|- ( ph -> R e. Ring )
ringinvnzdiv.x	|- ( ph -> X e. B )
ringinvnzdiv.a	|- ( ph -> E. a e. B ( a .x. X ) = .1. )
ringinvnzdiv.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
ringinvnzdiv	|- ( ph -> ( ( X .x. Y ) = .0. <-> Y = .0. ) )

Distinct variable groups:   X, a   .0. , a   .1. , a   .x. , a   ph, a   Y, a

Allowed substitution hints:   B( a)   R( a)

Proof of Theorem ringinvnzdiv

Step	Hyp	Ref	Expression
1		ringinvnzdiv.a	. . 3 |- ( ph -> E. a e. B ( a .x. X ) = .1. )
2		ringinvnzdiv.r	. . . . . . . . 9 |- ( ph -> R e. Ring )
3		ringinvnzdiv.y	. . . . . . . . 9 |- ( ph -> Y e. B )
4		ringinvnzdiv.b	. . . . . . . . . 10 |- B = ( Base ` R )
5		ringinvnzdiv.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
6		ringinvnzdiv.u	. . . . . . . . . 10 |- .1. = ( 1r ` R )
7	4, 5, 6	ringlidm 14035	. . . . . . . . 9 |- ( ( R e. Ring /\ Y e. B ) -> ( .1. .x. Y ) = Y )
8	2, 3, 7	syl2anc 411	. . . . . . . 8 |- ( ph -> ( .1. .x. Y ) = Y )
9	8	eqcomd 2237	. . . . . . 7 |- ( ph -> Y = ( .1. .x. Y ) )
10	9	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) /\ ( X .x. Y ) = .0. ) -> Y = ( .1. .x. Y ) )
11		oveq1 6024	. . . . . . . . . 10 |- ( .1. = ( a .x. X ) -> ( .1. .x. Y ) = ( ( a .x. X ) .x. Y ) )
12	11	eqcoms 2234	. . . . . . . . 9 |- ( ( a .x. X ) = .1. -> ( .1. .x. Y ) = ( ( a .x. X ) .x. Y ) )
13	12	adantl 277	. . . . . . . 8 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( .1. .x. Y ) = ( ( a .x. X ) .x. Y ) )
14	2	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ a e. B ) -> R e. Ring )
15		simpr 110	. . . . . . . . . . . 12 |- ( ( ph /\ a e. B ) -> a e. B )
16		ringinvnzdiv.x	. . . . . . . . . . . . 13 |- ( ph -> X e. B )
17	16	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ a e. B ) -> X e. B )
18	3	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ a e. B ) -> Y e. B )
19	15, 17, 18	3jca 1203	. . . . . . . . . . 11 |- ( ( ph /\ a e. B ) -> ( a e. B /\ X e. B /\ Y e. B ) )
20	14, 19	jca 306	. . . . . . . . . 10 |- ( ( ph /\ a e. B ) -> ( R e. Ring /\ ( a e. B /\ X e. B /\ Y e. B ) ) )
21	20	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( R e. Ring /\ ( a e. B /\ X e. B /\ Y e. B ) ) )
22	4, 5	ringass 14028	. . . . . . . . 9 |- ( ( R e. Ring /\ ( a e. B /\ X e. B /\ Y e. B ) ) -> ( ( a .x. X ) .x. Y ) = ( a .x. ( X .x. Y ) ) )
23	21, 22	syl 14	. . . . . . . 8 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( ( a .x. X ) .x. Y ) = ( a .x. ( X .x. Y ) ) )
24	13, 23	eqtrd 2264	. . . . . . 7 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( .1. .x. Y ) = ( a .x. ( X .x. Y ) ) )
25	24	adantr 276	. . . . . 6 |- ( ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) /\ ( X .x. Y ) = .0. ) -> ( .1. .x. Y ) = ( a .x. ( X .x. Y ) ) )
26		oveq2 6025	. . . . . . 7 |- ( ( X .x. Y ) = .0. -> ( a .x. ( X .x. Y ) ) = ( a .x. .0. ) )
27		ringinvnzdiv.z	. . . . . . . . . 10 |- .0. = ( 0g ` R )
28	4, 5, 27	ringrz 14056	. . . . . . . . 9 |- ( ( R e. Ring /\ a e. B ) -> ( a .x. .0. ) = .0. )
29	2, 28	sylan 283	. . . . . . . 8 |- ( ( ph /\ a e. B ) -> ( a .x. .0. ) = .0. )
30	29	adantr 276	. . . . . . 7 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( a .x. .0. ) = .0. )
31	26, 30	sylan9eqr 2286	. . . . . 6 |- ( ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) /\ ( X .x. Y ) = .0. ) -> ( a .x. ( X .x. Y ) ) = .0. )
32	10, 25, 31	3eqtrd 2268	. . . . 5 |- ( ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) /\ ( X .x. Y ) = .0. ) -> Y = .0. )
33	32	exp31 364	. . . 4 |- ( ( ph /\ a e. B ) -> ( ( a .x. X ) = .1. -> ( ( X .x. Y ) = .0. -> Y = .0. ) ) )
34	33	rexlimdva 2650	. . 3 |- ( ph -> ( E. a e. B ( a .x. X ) = .1. -> ( ( X .x. Y ) = .0. -> Y = .0. ) ) )
35	1, 34	mpd 13	. 2 |- ( ph -> ( ( X .x. Y ) = .0. -> Y = .0. ) )
36		oveq2 6025	. . . 4 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
37	4, 5, 27	ringrz 14056	. . . . 5 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )
38	2, 16, 37	syl2anc 411	. . . 4 |- ( ph -> ( X .x. .0. ) = .0. )
39	36, 38	sylan9eqr 2286	. . 3 |- ( ( ph /\ Y = .0. ) -> ( X .x. Y ) = .0. )
40	39	ex 115	. 2 |- ( ph -> ( Y = .0. -> ( X .x. Y ) = .0. ) )
41	35, 40	impbid 129	1 |- ( ph -> ( ( X .x. Y ) = .0. <-> Y = .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160   0gc0g 13338   1rcur 13971   Ringcrg 14008

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-mgp 13933  df-ur 13972  df-ring 14010

This theorem is referenced by: (None)