Theorem ringinvnz1ne0 14327

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 6083	. . . . 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 14322	. . . . . . 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 2251	. . . . . . . 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 6083	. . . . 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 14302	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = X )
17	3, 4, 5	ringrz 14322	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )
18	16, 17	eqeq12d 2253	. . . . . . . 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 415	. . . . . 6 |- ( ph -> ( ( X .x. .1. ) = ( X .x. .0. ) -> X = .0. ) )
21	20	ad2antrr 492	. . . . 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 2694	. 2 |- ( ph -> ( X = .0. <-> .1. = .0. ) )
26	25	necon3bid 2461	1 |- ( ph -> ( X =/= .0. <-> .1. =/= .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   ` cfv 5372  (class class class)co 6075   Basecbs 13330   .rcmulr 13409   0gc0g 13587   1rcur 14237   Ringcrg 14274

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-mgp 14195  df-ur 14238  df-ring 14276

This theorem is referenced by: (None)