Theorem ringinvnz1ne0 13299

Description: In a unital ring, a left invertible element is different from zero iff .1. =/= .0.. (Contributed by FL, 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. )

Assertion

Ref	Expression
ringinvnz1ne0	|- ( ph -> ( X =/= .0. <-> .1. =/= .0. ) )

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

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

Proof of Theorem ringinvnz1ne0

Step	Hyp	Ref	Expression
1		oveq2 5896	. . . . 5 |- ( X = .0. -> ( a .x. X ) = ( a .x. .0. ) )
2		ringinvnzdiv.r	. . . . . . 7 |- ( ph -> R e. Ring )
3		ringinvnzdiv.b	. . . . . . . 8 |- B = ( Base ` R )
4		ringinvnzdiv.t	. . . . . . . 8 |- .x. = ( .r ` R )
5		ringinvnzdiv.z	. . . . . . . 8 |- .0. = ( 0g ` R )
6	3, 4, 5	ringrz 13296	. . . . . . 7 |- ( ( R e. Ring /\ a e. B ) -> ( a .x. .0. ) = .0. )
7	2, 6	sylan 283	. . . . . 6 |- ( ( ph /\ a e. B ) -> ( a .x. .0. ) = .0. )
8		eqeq12 2200	. . . . . . . 8 |- ( ( ( a .x. X ) = .1. /\ ( a .x. .0. ) = .0. ) -> ( ( a .x. X ) = ( a .x. .0. ) <-> .1. = .0. ) )
9	8	biimpd 144	. . . . . . 7 |- ( ( ( a .x. X ) = .1. /\ ( a .x. .0. ) = .0. ) -> ( ( a .x. X ) = ( a .x. .0. ) -> .1. = .0. ) )
10	9	ex 115	. . . . . 6 |- ( ( a .x. X ) = .1. -> ( ( a .x. .0. ) = .0. -> ( ( a .x. X ) = ( a .x. .0. ) -> .1. = .0. ) ) )
11	7, 10	mpan9 281	. . . . 5 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( ( a .x. X ) = ( a .x. .0. ) -> .1. = .0. ) )
12	1, 11	syl5 32	. . . 4 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( X = .0. -> .1. = .0. ) )
13		oveq2 5896	. . . . 5 |- ( .1. = .0. -> ( X .x. .1. ) = ( X .x. .0. ) )
14		ringinvnzdiv.x	. . . . . . 7 |- ( ph -> X e. B )
15		ringinvnzdiv.u	. . . . . . . . . 10 |- .1. = ( 1r ` R )
16	3, 4, 15	ringridm 13276	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = X )
17	3, 4, 5	ringrz 13296	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )
18	16, 17	eqeq12d 2202	. . . . . . . 8 |- ( ( R e. Ring /\ X e. B ) -> ( ( X .x. .1. ) = ( X .x. .0. ) <-> X = .0. ) )
19	18	biimpd 144	. . . . . . 7 |- ( ( R e. Ring /\ X e. B ) -> ( ( X .x. .1. ) = ( X .x. .0. ) -> X = .0. ) )
20	2, 14, 19	syl2anc 411	. . . . . 6 |- ( ph -> ( ( X .x. .1. ) = ( X .x. .0. ) -> X = .0. ) )
21	20	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( ( X .x. .1. ) = ( X .x. .0. ) -> X = .0. ) )
22	13, 21	syl5 32	. . . 4 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( .1. = .0. -> X = .0. ) )
23	12, 22	impbid 129	. . 3 |- ( ( ( ph /\ a e. B ) /\ ( a .x. X ) = .1. ) -> ( X = .0. <-> .1. = .0. ) )
24		ringinvnzdiv.a	. . 3 |- ( ph -> E. a e. B ( a .x. X ) = .1. )
25	23, 24	r19.29a 2630	. 2 |- ( ph -> ( X = .0. <-> .1. = .0. ) )
26	25	necon3bid 2398	1 |- ( ph -> ( X =/= .0. <-> .1. =/= .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   =/= wne 2357   E.wrex 2466   ` cfv 5228  (class class class)co 5888   Basecbs 12476   .rcmulr 12552   0gc0g 12723   1rcur 13211   Ringcrg 13248

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-plusg 12564  df-mulr 12565  df-0g 12725  df-mgm 12794  df-sgrp 12827  df-mnd 12840  df-grp 12902  df-mgp 13173  df-ur 13212  df-ring 13250

This theorem is referenced by: (None)