Theorem ringinvnzdiv 13812

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 13785	. . . . . . . . 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 2211	. . . . . . 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 5951	. . . . . . . . . 10 |- ( .1. = ( a .x. X ) -> ( .1. .x. Y ) = ( ( a .x. X ) .x. Y ) )
12	11	eqcoms 2208	. . . . . . . . 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 1180	. . . . . . . . . . 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 13778	. . . . . . . . 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 2238	. . . . . . 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 5952	. . . . . . 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 13806	. . . . . . . . 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 2260	. . . . . 6 |- ( ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) /\ ( X .x. Y ) = .0. ) -> ( a .x. ( X .x. Y ) ) = .0. )
32	10, 25, 31	3eqtrd 2242	. . . . 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 2623	. . 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 5952	. . . 4 |- ( Y = .0. -> ( X .x. Y ) = ( X .x. .0. ) )
37	4, 5, 27	ringrz 13806	. . . . 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 2260	. . 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 981   = wceq 1373   e. wcel 2176   E.wrex 2485   ` cfv 5271  (class class class)co 5944   Basecbs 12832   .rcmulr 12910   0gc0g 13088   1rcur 13721   Ringcrg 13758

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-mgp 13683  df-ur 13722  df-ring 13760

This theorem is referenced by: (None)