Theorem ringinvnz1ne0 14143

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 6036	. . . . 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 14138	. . . . . . 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 2244	. . . . . . . 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 6036	. . . . 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 14118	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .1. ) = X )
17	3, 4, 5	ringrz 14138	. . . . . . . . 9 |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )
18	16, 17	eqeq12d 2246	. . . . . . . 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 2677	. 2 |- ( ph -> ( X = .0. <-> .1. = .0. ) )
26	25	necon3bid 2444	1 |- ( ph -> ( X =/= .0. <-> .1. =/= .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   =/= wne 2403   E.wrex 2512   ` cfv 5333  (class class class)co 6028   Basecbs 13162   .rcmulr 13241   0gc0g 13419   1rcur 14053   Ringcrg 14090

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-mgp 14015  df-ur 14054  df-ring 14092

This theorem is referenced by: (None)