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Theorem zerdivemp1x 31640
Description: In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 25850 by Frederic Line. (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
StepHypRef Expression
1 oveq2 6286 . . . . . . 7  |-  ( ( A H B )  =  Z  ->  (
a H ( A H B ) )  =  ( a H Z ) )
2 simpl1 1000 . . . . . . . . . 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 1003 . . . . . . . . . 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 1005 . . . . . . . . . 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 1002 . . . . . . . . . 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
96, 7, 8rngoass 25803 . . . . . . . . . 10  |-  ( ( R  e.  RingOps  /\  (
a  e.  X  /\  A  e.  X  /\  B  e.  X )
)  ->  ( (
a H A ) H B )  =  ( a H ( A H B ) ) )
102, 3, 4, 5, 9syl13anc 1232 . . . . . . . . 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 2428 . . . . . . . . . . . . 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 ) )
1211ex 432 . . . . . . . . . . . 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 oveq1 6285 . . . . . . . . . . . . . . . 16  |-  ( ( a H A )  =  U  ->  (
( a H A ) H B )  =  ( U H B ) )
14 eqtr 2428 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( U H B )  =  ( ( a H A ) H B )  /\  ( ( a H A ) H B )  =  ( a H Z ) )  ->  ( U H B )  =  ( a H Z ) )
15 zerdivempx.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  Z  =  (GId `  G )
1615, 8, 6, 7rngorz 25818 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
17163adant3 1017 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  (
a H Z )  =  Z )
186rneqi 5050 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ran  G  =  ran  ( 1st `  R
)
198, 18eqtri 2431 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  X  =  ran  ( 1st `  R
)
20 zerdivempx.5 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  U  =  (GId `  H )
217, 19, 20rngolidm 25840 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  ( U H B )  =  B )
22213adant2 1016 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( U H B )  =  B )
23 simp1 997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  ( a H Z ) )
24 simp2 998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  B )
25 simp3 999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( a H Z )  =  Z )
2623, 24, 253eqtr3d 2451 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  ->  B  =  Z )
2726a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( A  e.  X  ->  B  =  Z ) )
28273exp 1196 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( U H B )  =  ( a H Z )  ->  (
( U H B )  =  B  -> 
( ( a H Z )  =  Z  ->  ( A  e.  X  ->  B  =  Z ) ) ) )
2928com14 88 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( A  e.  X  ->  (
( U H B )  =  B  -> 
( ( a H Z )  =  Z  ->  ( ( U H B )  =  ( a H Z )  ->  B  =  Z ) ) ) )
3029com13 80 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( a H Z )  =  Z  ->  (
( U H B )  =  B  -> 
( A  e.  X  ->  ( ( U H B )  =  ( a H Z )  ->  B  =  Z ) ) ) )
3117, 22, 30sylc 59 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( A  e.  X  ->  ( ( U H B )  =  ( a H Z )  ->  B  =  Z )
) )
32313exp 1196 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( R  e.  RingOps  ->  ( a  e.  X  ->  ( B  e.  X  ->  ( A  e.  X  ->  (
( U H B )  =  ( a H Z )  ->  B  =  Z )
) ) ) )
3332com15 93 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U H B )  =  ( a H Z )  ->  (
a  e.  X  -> 
( B  e.  X  ->  ( A  e.  X  ->  ( R  e.  RingOps  ->  B  =  Z )
) ) ) )
3433com24 87 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U H B )  =  ( a H Z )  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( a  e.  X  ->  ( R  e.  RingOps  ->  B  =  Z )
) ) ) )
3514, 34syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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 ) ) ) ) )
3635ex 432 . . . . . . . . . . . . . . . . . 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 )
) ) ) ) )
3736eqcoms 2414 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 )
) ) ) ) )
3837com25 91 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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
) ) ) ) ) )
3913, 38syl 17 . . . . . . . . . . . . . . 15  |-  ( ( a H A )  =  U  ->  (
a  e.  X  -> 
( A  e.  X  ->  ( B  e.  X  ->  ( ( ( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps 
->  B  =  Z
) ) ) ) ) )
4039com12 29 . . . . . . . . . . . . . 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
) ) ) ) ) )
41403imp 1191 . . . . . . . . . . . . 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
) ) ) )
4241com13 80 . . . . . . . . . . . 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 )
) ) )
4312, 42syl6 31 . . . . . . . . . . 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 ) ) ) ) )
4443com15 93 . . . . . . . . . 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 ) ) ) ) )
45443imp1 1210 . . . . . . . . 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 ) )
4610, 45mpd 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 )
47463exp1 1213 . . . . . . 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 ) ) ) )
481, 47syl5com 28 . . . . . 6  |-  ( ( A H B )  =  Z  ->  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  ->  B  =  Z ) ) ) )
4948com14 88 . . . . 5  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  ->  ( R  e.  RingOps  -> 
( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) ) )
50493exp 1196 . . . 4  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  X  ->  ( R  e.  RingOps  -> 
( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) ) ) ) )
5150rexlimiv 2890 . . 3  |-  ( E. a  e.  X  ( a H A )  =  U  ->  ( A  e.  X  ->  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z )
) ) ) )
5251com13 80 . 2  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( E. a  e.  X  (
a H A )  =  U  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z )
) ) ) )
53523imp 1191 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2755   ran crn 4824   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783  GIdcgi 25603   RingOpscrngo 25791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577  df-riota 6240  df-ov 6281  df-1st 6784  df-2nd 6785  df-grpo 25607  df-gid 25608  df-ablo 25698  df-ass 25729  df-exid 25731  df-mgmOLD 25735  df-sgrOLD 25747  df-mndo 25754  df-rngo 25792
This theorem is referenced by:  isdrngo2  31643
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