Theorem zerdivemp1x 33746

Description: In a unitary ring a left invertible element is not a zero divisor. See also ringinvnzdiv 18593. (Contributed by Jeff Madsen, 18-Apr-2010.)

Hypotheses

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

Assertion

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

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

Allowed substitution hints:   U( a)   G( a)

Proof of Theorem zerdivemp1x

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . 7 |- ( ( A H B ) = Z -> ( a H ( A H B ) ) = ( a H Z ) )
2		simpl1 1064	. . . . . . . . . 10 |- ( ( ( R e. RingOps /\ ( a H ( A H B ) ) = ( a H Z ) /\ B e. X ) /\ ( a e. X /\ ( a H A ) = U /\ A e. X ) ) -> R e. RingOps )
3		simpr1 1067	. . . . . . . . . 10 |- ( ( ( R e. RingOps /\ ( a H ( A H B ) ) = ( a H Z ) /\ B e. X ) /\ ( a e. X /\ ( a H A ) = U /\ A e. X ) ) -> a e. X )
4		simpr3 1069	. . . . . . . . . 10 |- ( ( ( R e. RingOps /\ ( a H ( A H B ) ) = ( a H Z ) /\ B e. X ) /\ ( a e. X /\ ( a H A ) = U /\ A e. X ) ) -> A e. X )
5		simpl3 1066	. . . . . . . . . 10 |- ( ( ( R e. RingOps /\ ( a H ( A H B ) ) = ( a H Z ) /\ B e. X ) /\ ( a e. X /\ ( a H A ) = U /\ A e. X ) ) -> B e. X )
6		zerdivempx.1	. . . . . . . . . . 11 |- G = ( 1st ` R )
7		zerdivempx.2	. . . . . . . . . . 11 |- H = ( 2nd ` R )
8		zerdivempx.4	. . . . . . . . . . 11 |- X = ran G
9	6, 7, 8	rngoass 33705	. . . . . . . . . 10 |- ( ( R e. RingOps /\ ( a e. X /\ A e. X /\ B e. X ) ) -> ( ( a H A ) H B ) = ( a H ( A H B ) ) )
10	2, 3, 4, 5, 9	syl13anc 1328	. . . . . . . . 9 |- ( ( ( R e. RingOps /\ ( a H ( A H B ) ) = ( a H Z ) /\ B e. X ) /\ ( a e. X /\ ( a H A ) = U /\ A e. X ) ) -> ( ( a H A ) H B ) = ( a H ( A H B ) ) )
11		eqtr 2641	. . . . . . . . . . . . 13 |- ( ( ( ( a H A ) H B ) = ( a H ( A H B ) ) /\ ( a H ( A H B ) ) = ( a H Z ) ) -> ( ( a H A ) H B ) = ( a H Z ) )
12	11	ex 450	. . . . . . . . . . . 12 |- ( ( ( a H A ) H B ) = ( a H ( A H B ) ) -> ( ( a H ( A H B ) ) = ( a H Z ) -> ( ( a H A ) H B ) = ( a H Z ) ) )
13		eqtr 2641	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( U H B ) = ( ( a H A ) H B ) /\ ( ( a H A ) H B ) = ( a H Z ) ) -> ( U H B ) = ( a H Z ) )
14		zerdivempx.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- Z = (GId ` G )
15	14, 8, 6, 7	rngorz 33722	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( R e. RingOps /\ a e. X ) -> ( a H Z ) = Z )
16	15	3adant3 1081	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( R e. RingOps /\ a e. X /\ B e. X ) -> ( a H Z ) = Z )
17	6	rneqi 5352	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ran G = ran ( 1st ` R )
18	8, 17	eqtri 2644	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- X = ran ( 1st ` R )
19		zerdivempx.5	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- U = (GId ` H )
20	7, 18, 19	rngolidm 33736	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( R e. RingOps /\ B e. X ) -> ( U H B ) = B )
21	20	3adant2 1080	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( R e. RingOps /\ a e. X /\ B e. X ) -> ( U H B ) = B )
22		simp1 1061	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( U H B ) = ( a H Z ) /\ ( U H B ) = B /\ ( a H Z ) = Z ) -> ( U H B ) = ( a H Z ) )
23		simp2 1062	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( U H B ) = ( a H Z ) /\ ( U H B ) = B /\ ( a H Z ) = Z ) -> ( U H B ) = B )
24		simp3 1063	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( U H B ) = ( a H Z ) /\ ( U H B ) = B /\ ( a H Z ) = Z ) -> ( a H Z ) = Z )
25	22, 23, 24	3eqtr3d 2664	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( U H B ) = ( a H Z ) /\ ( U H B ) = B /\ ( a H Z ) = Z ) -> B = Z )
26	25	a1d 25	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( U H B ) = ( a H Z ) /\ ( U H B ) = B /\ ( a H Z ) = Z ) -> ( A e. X -> B = Z ) )
27	26	3exp 1264	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( U H B ) = ( a H Z ) -> ( ( U H B ) = B -> ( ( a H Z ) = Z -> ( A e. X -> B = Z ) ) ) )
28	27	com14 96	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( A e. X -> ( ( U H B ) = B -> ( ( a H Z ) = Z -> ( ( U H B ) = ( a H Z ) -> B = Z ) ) ) )
29	28	com13 88	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( a H Z ) = Z -> ( ( U H B ) = B -> ( A e. X -> ( ( U H B ) = ( a H Z ) -> B = Z ) ) ) )
30	16, 21, 29	sylc 65	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( R e. RingOps /\ a e. X /\ B e. X ) -> ( A e. X -> ( ( U H B ) = ( a H Z ) -> B = Z ) ) )
31	30	3exp 1264	. . . . . . . . . . . . . . . . . . . . 21 |- ( R e. RingOps -> ( a e. X -> ( B e. X -> ( A e. X -> ( ( U H B ) = ( a H Z ) -> B = Z ) ) ) ) )
32	31	com15 101	. . . . . . . . . . . . . . . . . . . 20 |- ( ( U H B ) = ( a H Z ) -> ( a e. X -> ( B e. X -> ( A e. X -> ( R e. RingOps -> B = Z ) ) ) ) )
33	32	com24 95	. . . . . . . . . . . . . . . . . . 19 |- ( ( U H B ) = ( a H Z ) -> ( A e. X -> ( B e. X -> ( a e. X -> ( R e. RingOps -> B = Z ) ) ) ) )
34	13, 33	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( U H B ) = ( ( a H A ) H B ) /\ ( ( a H A ) H B ) = ( a H Z ) ) -> ( A e. X -> ( B e. X -> ( a e. X -> ( R e. RingOps -> B = Z ) ) ) ) )
35	34	ex 450	. . . . . . . . . . . . . . . . 17 |- ( ( U H B ) = ( ( a H A ) H B ) -> ( ( ( a H A ) H B ) = ( a H Z ) -> ( A e. X -> ( B e. X -> ( a e. X -> ( R e. RingOps -> B = Z ) ) ) ) ) )
36	35	eqcoms 2630	. . . . . . . . . . . . . . . 16 |- ( ( ( a H A ) H B ) = ( U H B ) -> ( ( ( a H A ) H B ) = ( a H Z ) -> ( A e. X -> ( B e. X -> ( a e. X -> ( R e. RingOps -> B = Z ) ) ) ) ) )
37	36	com25 99	. . . . . . . . . . . . . . 15 |- ( ( ( a H A ) H B ) = ( U H B ) -> ( a e. X -> ( A e. X -> ( B e. X -> ( ( ( a H A ) H B ) = ( a H Z ) -> ( R e. RingOps -> B = Z ) ) ) ) ) )
38		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( ( a H A ) = U -> ( ( a H A ) H B ) = ( U H B ) )
39	37, 38	syl11 33	. . . . . . . . . . . . . 14 |- ( a e. X -> ( ( a H A ) = U -> ( A e. X -> ( B e. X -> ( ( ( a H A ) H B ) = ( a H Z ) -> ( R e. RingOps -> B = Z ) ) ) ) ) )
40	39	3imp 1256	. . . . . . . . . . . . 13 |- ( ( a e. X /\ ( a H A ) = U /\ A e. X ) -> ( B e. X -> ( ( ( a H A ) H B ) = ( a H Z ) -> ( R e. RingOps -> B = Z ) ) ) )
41	40	com13 88	. . . . . . . . . . . 12 |- ( ( ( a H A ) H B ) = ( a H Z ) -> ( B e. X -> ( ( a e. X /\ ( a H A ) = U /\ A e. X ) -> ( R e. RingOps -> B = Z ) ) ) )
42	12, 41	syl6 35	. . . . . . . . . . 11 |- ( ( ( a H A ) H B ) = ( a H ( A H B ) ) -> ( ( a H ( A H B ) ) = ( a H Z ) -> ( B e. X -> ( ( a e. X /\ ( a H A ) = U /\ A e. X ) -> ( R e. RingOps -> B = Z ) ) ) ) )
43	42	com15 101	. . . . . . . . . 10 |- ( R e. RingOps -> ( ( a H ( A H B ) ) = ( a H Z ) -> ( B e. X -> ( ( a e. X /\ ( a H A ) = U /\ A e. X ) -> ( ( ( a H A ) H B ) = ( a H ( A H B ) ) -> B = Z ) ) ) ) )
44	43	3imp1 1280	. . . . . . . . 9 |- ( ( ( R e. RingOps /\ ( a H ( A H B ) ) = ( a H Z ) /\ B e. X ) /\ ( a e. X /\ ( a H A ) = U /\ A e. X ) ) -> ( ( ( a H A ) H B ) = ( a H ( A H B ) ) -> B = Z ) )
45	10, 44	mpd 15	. . . . . . . 8 |- ( ( ( R e. RingOps /\ ( a H ( A H B ) ) = ( a H Z ) /\ B e. X ) /\ ( a e. X /\ ( a H A ) = U /\ A e. X ) ) -> B = Z )
46	45	3exp1 1283	. . . . . . 7 |- ( R e. RingOps -> ( ( a H ( A H B ) ) = ( a H Z ) -> ( B e. X -> ( ( a e. X /\ ( a H A ) = U /\ A e. X ) -> B = Z ) ) ) )
47	1, 46	syl5com 31	. . . . . 6 |- ( ( A H B ) = Z -> ( R e. RingOps -> ( B e. X -> ( ( a e. X /\ ( a H A ) = U /\ A e. X ) -> B = Z ) ) ) )
48	47	com14 96	. . . . 5 |- ( ( a e. X /\ ( a H A ) = U /\ A e. X ) -> ( R e. RingOps -> ( B e. X -> ( ( A H B ) = Z -> B = Z ) ) ) )
49	48	3exp 1264	. . . 4 |- ( a e. X -> ( ( a H A ) = U -> ( A e. X -> ( R e. RingOps -> ( B e. X -> ( ( A H B ) = Z -> B = Z ) ) ) ) ) )
50	49	rexlimiv 3027	. . 3 |- ( E. a e. X ( a H A ) = U -> ( A e. X -> ( R e. RingOps -> ( B e. X -> ( ( A H B ) = Z -> B = Z ) ) ) ) )
51	50	com13 88	. 2 |- ( R e. RingOps -> ( A e. X -> ( E. a e. X ( a H A ) = U -> ( B e. X -> ( ( A H B ) = Z -> B = Z ) ) ) ) )
52	51	3imp 1256	1 |- ( ( R e. RingOps /\ A e. X /\ E. a e. X ( a H A ) = U ) -> ( B e. X -> ( ( A H B ) = Z -> B = Z ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694

This theorem is referenced by:  isdrngo2  33757