Theorem ringinvnzdiv 14194

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 14167	. . . . . . . . 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 2238	. . . . . . 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 6057	. . . . . . . . . 10 |- ( .1. = ( a .x. X ) -> ( .1. .x. Y ) = ( ( a .x. X ) .x. Y ) )
12	11	eqcoms 2235	. . . . . . . . 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 1204	. . . . . . . . . . 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 14160	. . . . . . . . 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 2265	. . . . . . 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 6058	. . . . . . 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 14188	. . . . . . . . 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 2287	. . . . . 6 |- ( ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) /\ ( X .x. Y ) = .0. ) -> ( a .x. ( X .x. Y ) ) = .0. )
32	10, 25, 31	3eqtrd 2269	. . . . 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 2660	. . 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 6058	. . . 4 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
37	4, 5, 27	ringrz 14188	. . . . 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 2287	. . 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 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   ` cfv 5352  (class class class)co 6050   Basecbs 13212   .rcmulr 13291   0gc0g 13469   1rcur 14103   Ringcrg 14140

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-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

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 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-mgp 14065  df-ur 14104  df-ring 14142

This theorem is referenced by: (None)