Theorem ringinvnzdiv 13019

Description: In a unitary 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 12999	. . . . . . . . 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 2181	. . . . . . 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 5872	. . . . . . . . . 10 |- ( .1. = ( a .x. X ) -> ( .1. .x. Y ) = ( ( a .x. X ) .x. Y ) )
12	11	eqcoms 2178	. . . . . . . . 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 1177	. . . . . . . . . . 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 12992	. . . . . . . . 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 2208	. . . . . . 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 5873	. . . . . . 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 13015	. . . . . . . . 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 2230	. . . . . 6 |- ( ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) /\ ( X .x. Y ) = .0. ) -> ( a .x. ( X .x. Y ) ) = .0. )
32	10, 25, 31	3eqtrd 2212	. . . . 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 2592	. . 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 5873	. . . 4 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
37	4, 5, 27	ringrz 13015	. . . . 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 2230	. . 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 978   = wceq 1353   e. wcel 2146   E.wrex 2454   ` cfv 5208  (class class class)co 5865   Basecbs 12427   .rcmulr 12492   0gc0g 12625   1rcur 12935   Ringcrg 12972

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12430  df-slot 12431  df-base 12433  df-sets 12434  df-plusg 12504  df-mulr 12505  df-0g 12627  df-mgm 12639  df-sgrp 12672  df-mnd 12682  df-grp 12740  df-mgp 12926  df-ur 12936  df-ring 12974

This theorem is referenced by: (None)