Theorem rngridfz 14786

Description: In a unitary ring a left invertible element is different from zero iff 1 =/= 0.

Hypotheses

Ref	Expression
zerdivemp.1	|- G = (1st` R)
zerdivemp.2	|- H = (2nd` R)
zerdivemp.3	|- Z = (Id` G)
zerdivemp.4	|- X = ran G
zerdivemp.5	|- U = (Id` H)

Assertion

Ref	Expression
rngridfz	|- ((R e. Ring /\ A e. X /\ E.a e. X (aHA) = U) -> (A =/= Z <-> U =/= Z))

Distinct variable groups:   A,a   R,a   U,a   X,a   Z,a

Proof of Theorem rngridfz

Step	Hyp	Ref	Expression
1		eqeq12 1896	. . . . . . . . . . . . . . . 16 |- (((aHA) = U /\ (aHZ) = Z) -> ((aHA) = (aHZ) <-> U = Z))
2	1	biimpd 170	. . . . . . . . . . . . . . 15 |- (((aHA) = U /\ (aHZ) = Z) -> ((aHA) = (aHZ) -> U = Z))
3	2	ex 402	. . . . . . . . . . . . . 14 |- ((aHA) = U -> ((aHZ) = Z -> ((aHA) = (aHZ) -> U = Z)))
4		zerdivemp.3	. . . . . . . . . . . . . . 15 |- Z = (Id` G)
5		zerdivemp.4	. . . . . . . . . . . . . . 15 |- X = ran G
6		zerdivemp.1	. . . . . . . . . . . . . . 15 |- G = (1st` R)
7		zerdivemp.2	. . . . . . . . . . . . . . 15 |- H = (2nd` R)
8	4, 5, 6, 7	ringrz 9488	. . . . . . . . . . . . . 14 |- ((R e. Ring /\ a e. X) -> (aHZ) = Z)
9	3, 8	syl5com 63	. . . . . . . . . . . . 13 |- ((R e. Ring /\ a e. X) -> ((aHA) = U -> ((aHA) = (aHZ) -> U = Z)))
10	9	ex 402	. . . . . . . . . . . 12 |- (R e. Ring -> (a e. X -> ((aHA) = U -> ((aHA) = (aHZ) -> U = Z))))
11	10	com3l 38	. . . . . . . . . . 11 |- (a e. X -> ((aHA) = U -> (R e. Ring -> ((aHA) = (aHZ) -> U = Z))))
12	11	imp 377	. . . . . . . . . 10 |- ((a e. X /\ (aHA) = U) -> (R e. Ring -> ((aHA) = (aHZ) -> U = Z)))
13	12	3adant3 896	. . . . . . . . 9 |- ((a e. X /\ (aHA) = U /\ A e. X) -> (R e. Ring -> ((aHA) = (aHZ) -> U = Z)))
14	13	imp 377	. . . . . . . 8 |- (((a e. X /\ (aHA) = U /\ A e. X) /\ R e. Ring) -> ((aHA) = (aHZ) -> U = Z))
15		opreq2 4890	. . . . . . . 8 |- (A = Z -> (aHA) = (aHZ))
16	14, 15	syl5 20	. . . . . . 7 |- (((a e. X /\ (aHA) = U /\ A e. X) /\ R e. Ring) -> (A = Z -> U = Z))
17	6	rneqi 4187	. . . . . . . . . . . . . 14 |- ran G = ran (1st` R)
18	5, 17	eqtri 1908	. . . . . . . . . . . . 13 |- X = ran (1st` R)
19		zerdivemp.5	. . . . . . . . . . . . 13 |- U = (Id` H)
20	7, 18, 19	ringridm 10411	. . . . . . . . . . . 12 |- ((R e. Ring /\ A e. X) -> (AHU) = A)
21	4, 5, 6, 7	ringrz 9488	. . . . . . . . . . . 12 |- ((R e. Ring /\ A e. X) -> (AHZ) = Z)
22		eqeq12 1896	. . . . . . . . . . . . 13 |- (((AHU) = A /\ (AHZ) = Z) -> ((AHU) = (AHZ) <-> A = Z))
23	22	biimpd 170	. . . . . . . . . . . 12 |- (((AHU) = A /\ (AHZ) = Z) -> ((AHU) = (AHZ) -> A = Z))
24	20, 21, 23	syl11anc 524	. . . . . . . . . . 11 |- ((R e. Ring /\ A e. X) -> ((AHU) = (AHZ) -> A = Z))
25	24	expcom 403	. . . . . . . . . 10 |- (A e. X -> (R e. Ring -> ((AHU) = (AHZ) -> A = Z)))
26	25	3ad2ant3 899	. . . . . . . . 9 |- ((a e. X /\ (aHA) = U /\ A e. X) -> (R e. Ring -> ((AHU) = (AHZ) -> A = Z)))
27	26	imp 377	. . . . . . . 8 |- (((a e. X /\ (aHA) = U /\ A e. X) /\ R e. Ring) -> ((AHU) = (AHZ) -> A = Z))
28		opreq2 4890	. . . . . . . 8 |- (U = Z -> (AHU) = (AHZ))
29	27, 28	syl5 20	. . . . . . 7 |- (((a e. X /\ (aHA) = U /\ A e. X) /\ R e. Ring) -> (U = Z -> A = Z))
30	16, 29	impbid 574	. . . . . 6 |- (((a e. X /\ (aHA) = U /\ A e. X) /\ R e. Ring) -> (A = Z <-> U = Z))
31	30	3exp1 1084	. . . . 5 |- (a e. X -> ((aHA) = U -> (A e. X -> (R e. Ring -> (A = Z <-> U = Z)))))
32	31	r19.23aiv 2211	. . . 4 |- (E.a e. X (aHA) = U -> (A e. X -> (R e. Ring -> (A = Z <-> U = Z))))
33	32	com13 37	. . 3 |- (R e. Ring -> (A e. X -> (E.a e. X (aHA) = U -> (A = Z <-> U = Z))))
34	33	3imp 1061	. 2 |- ((R e. Ring /\ A e. X /\ E.a e. X (aHA) = U) -> (A = Z <-> U = Z))
35	34	necon3bid 2035	1 |- ((R e. Ring /\ A e. X /\ E.a e. X (aHA) = U) -> (A =/= Z <-> U =/= Z))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-abl 9408  df-ring 9464  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385

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