Theorem zerdivemp1x 16108

Description: In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 14785 by Frederic Line.

Hypotheses

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

Assertion

Ref	Expression
zerdivemp1x	|- ((R e. Ring /\ A e. X /\ E.a e. X (aHA) = U) -> (B e. X -> ((AHB) = Z -> B = Z)))

Distinct variable groups:   A,a   B,a   H,a   R,a   X,a   Z,a

Proof of Theorem zerdivemp1x

Step	Hyp	Ref	Expression
1		simpl1 879	. . . . . . . . . . 11 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> R e. Ring)
2		simpr1 882	. . . . . . . . . . . 12 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> a e. X)
3		simpr3 884	. . . . . . . . . . . 12 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> A e. X)
4		simpl3 881	. . . . . . . . . . . 12 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> B e. X)
5	2, 3, 4	3jca 1050	. . . . . . . . . . 11 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> (a e. X /\ A e. X /\ B e. X))
6	1, 5	jca 310	. . . . . . . . . 10 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> (R e. Ring /\ (a e. X /\ A e. X /\ B e. X)))
7		zerdivempx.1	. . . . . . . . . . 11 |- G = (1st` R)
8		zerdivempx.2	. . . . . . . . . . 11 |- H = (2nd` R)
9		zerdivempx.4	. . . . . . . . . . 11 |- X = ran G
10	7, 8, 9	ringass 9473	. . . . . . . . . 10 |- ((R e. Ring /\ (a e. X /\ A e. X /\ B e. X)) -> ((aHA)HB) = (aH(AHB)))
11		eqtr 1904	. . . . . . . . . . . . . . 15 |- ((((aHA)HB) = (aH(AHB)) /\ (aH(AHB)) = (aHZ)) -> ((aHA)HB) = (aHZ))
12	11	ex 402	. . . . . . . . . . . . . 14 |- (((aHA)HB) = (aH(AHB)) -> ((aH(AHB)) = (aHZ) -> ((aHA)HB) = (aHZ)))
13		opreq1 4889	. . . . . . . . . . . . . . . . . 18 |- ((aHA) = U -> ((aHA)HB) = (UHB))
14		eqtr 1904	. . . . . . . . . . . . . . . . . . . . . 22 |- (((UHB) = ((aHA)HB) /\ ((aHA)HB) = (aHZ)) -> (UHB) = (aHZ))
15		zerdivempx.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- Z = (Id` G)
16	15, 9, 7, 8	ringrz 9488	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((R e. Ring /\ a e. X) -> (aHZ) = Z)
17	16	3adant3 896	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((R e. Ring /\ a e. X /\ B e. X) -> (aHZ) = Z)
18	7	rneqi 4187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ran G = ran (1st` R)
19	9, 18	eqtri 1908	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- X = ran (1st` R)
20		zerdivempx.5	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- U = (Id` H)
21	8, 19, 20	ringlidm 10410	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((R e. Ring /\ B e. X) -> (UHB) = B)
22	21	3adant2 895	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((R e. Ring /\ a e. X /\ B e. X) -> (UHB) = B)
23		simp1 876	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (((UHB) = (aHZ) /\ (UHB) = B /\ (aHZ) = Z) -> (UHB) = (aHZ))
24		simp2 877	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (((UHB) = (aHZ) /\ (UHB) = B /\ (aHZ) = Z) -> (UHB) = B)
25		simp3 878	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (((UHB) = (aHZ) /\ (UHB) = B /\ (aHZ) = Z) -> (aHZ) = Z)
26	23, 24, 25	3eqtr3d 1934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (((UHB) = (aHZ) /\ (UHB) = B /\ (aHZ) = Z) -> B = Z)
27	26	a1d 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (((UHB) = (aHZ) /\ (UHB) = B /\ (aHZ) = Z) -> (A e. X -> B = Z))
28	27	3exp 1066	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((UHB) = (aHZ) -> ((UHB) = B -> ((aHZ) = Z -> (A e. X -> B = Z))))
29	28	com14 42	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (A e. X -> ((UHB) = B -> ((aHZ) = Z -> ((UHB) = (aHZ) -> B = Z))))
30	29	com13 37	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((aHZ) = Z -> ((UHB) = B -> (A e. X -> ((UHB) = (aHZ) -> B = Z))))
31	17, 22, 30	sylc 83	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((R e. Ring /\ a e. X /\ B e. X) -> (A e. X -> ((UHB) = (aHZ) -> B = Z)))
32	31	3exp 1066	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (R e. Ring -> (a e. X -> (B e. X -> (A e. X -> ((UHB) = (aHZ) -> B = Z)))))
33	32	com15 49	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((UHB) = (aHZ) -> (a e. X -> (B e. X -> (A e. X -> (R e. Ring -> B = Z)))))
34	33	com24 41	. . . . . . . . . . . . . . . . . . . . . 22 |- ((UHB) = (aHZ) -> (A e. X -> (B e. X -> (a e. X -> (R e. Ring -> B = Z)))))
35	14, 34	syl 12	. . . . . . . . . . . . . . . . . . . . 21 |- (((UHB) = ((aHA)HB) /\ ((aHA)HB) = (aHZ)) -> (A e. X -> (B e. X -> (a e. X -> (R e. Ring -> B = Z)))))
36	35	ex 402	. . . . . . . . . . . . . . . . . . . 20 |- ((UHB) = ((aHA)HB) -> (((aHA)HB) = (aHZ) -> (A e. X -> (B e. X -> (a e. X -> (R e. Ring -> B = Z))))))
37	36	eqcoms 1887	. . . . . . . . . . . . . . . . . . 19 |- (((aHA)HB) = (UHB) -> (((aHA)HB) = (aHZ) -> (A e. X -> (B e. X -> (a e. X -> (R e. Ring -> B = Z))))))
38	37	com25 48	. . . . . . . . . . . . . . . . . 18 |- (((aHA)HB) = (UHB) -> (a e. X -> (A e. X -> (B e. X -> (((aHA)HB) = (aHZ) -> (R e. Ring -> B = Z))))))
39	13, 38	syl 12	. . . . . . . . . . . . . . . . 17 |- ((aHA) = U -> (a e. X -> (A e. X -> (B e. X -> (((aHA)HB) = (aHZ) -> (R e. Ring -> B = Z))))))
40	39	com12 14	. . . . . . . . . . . . . . . 16 |- (a e. X -> ((aHA) = U -> (A e. X -> (B e. X -> (((aHA)HB) = (aHZ) -> (R e. Ring -> B = Z))))))
41	40	3imp 1061	. . . . . . . . . . . . . . 15 |- ((a e. X /\ (aHA) = U /\ A e. X) -> (B e. X -> (((aHA)HB) = (aHZ) -> (R e. Ring -> B = Z))))
42	41	com13 37	. . . . . . . . . . . . . 14 |- (((aHA)HB) = (aHZ) -> (B e. X -> ((a e. X /\ (aHA) = U /\ A e. X) -> (R e. Ring -> B = Z))))
43	12, 42	syl6 25	. . . . . . . . . . . . 13 |- (((aHA)HB) = (aH(AHB)) -> ((aH(AHB)) = (aHZ) -> (B e. X -> ((a e. X /\ (aHA) = U /\ A e. X) -> (R e. Ring -> B = Z)))))
44	43	com15 49	. . . . . . . . . . . 12 |- (R e. Ring -> ((aH(AHB)) = (aHZ) -> (B e. X -> ((a e. X /\ (aHA) = U /\ A e. X) -> (((aHA)HB) = (aH(AHB)) -> B = Z)))))
45	44	3imp1 1081	. . . . . . . . . . 11 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> (((aHA)HB) = (aH(AHB)) -> B = Z))
46	45	com12 14	. . . . . . . . . 10 |- (((aHA)HB) = (aH(AHB)) -> (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> B = Z))
47	6, 10, 46	3syl 24	. . . . . . . . 9 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> B = Z))
48	47	pm2.43i 78	. . . . . . . 8 |- (((R e. Ring /\ (aH(AHB)) = (aHZ) /\ B e. X) /\ (a e. X /\ (aHA) = U /\ A e. X)) -> B = Z)
49	48	3exp1 1084	. . . . . . 7 |- (R e. Ring -> ((aH(AHB)) = (aHZ) -> (B e. X -> ((a e. X /\ (aHA) = U /\ A e. X) -> B = Z))))
50		opreq2 4890	. . . . . . 7 |- ((AHB) = Z -> (aH(AHB)) = (aHZ))
51	49, 50	syl5com 63	. . . . . 6 |- ((AHB) = Z -> (R e. Ring -> (B e. X -> ((a e. X /\ (aHA) = U /\ A e. X) -> B = Z))))
52	51	com14 42	. . . . 5 |- ((a e. X /\ (aHA) = U /\ A e. X) -> (R e. Ring -> (B e. X -> ((AHB) = Z -> B = Z))))
53	52	3exp 1066	. . . 4 |- (a e. X -> ((aHA) = U -> (A e. X -> (R e. Ring -> (B e. X -> ((AHB) = Z -> B = Z))))))
54	53	r19.23aiv 2211	. . 3 |- (E.a e. X (aHA) = U -> (A e. X -> (R e. Ring -> (B e. X -> ((AHB) = Z -> B = Z)))))
55	54	com13 37	. 2 |- (R e. Ring -> (A e. X -> (E.a e. X (aHA) = U -> (B e. X -> ((AHB) = Z -> B = Z)))))
56	55	3imp 1061	1 |- ((R e. Ring /\ A e. X /\ E.a e. X (aHA) = U) -> (B e. X -> ((AHB) = Z -> B = Z)))

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

This theorem is referenced by:  isdivrng2 16111

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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