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Theorem zerdivemp1x 28787
Description: In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 23943 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 6120 . . . . . . 7  |-  ( ( A H B )  =  Z  ->  (
a H ( A H B ) )  =  ( a H Z ) )
2 simpl1 991 . . . . . . . . . 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 994 . . . . . . . . . 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 996 . . . . . . . . . 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 993 . . . . . . . . . 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 23896 . . . . . . . . . 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 1220 . . . . . . . . 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 2460 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 6119 . . . . . . . . . . . . . . . 16  |-  ( ( a H A )  =  U  ->  (
( a H A ) H B )  =  ( U H B ) )
14 eqtr 2460 . . . . . . . . . . . . . . . . . . . 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 23911 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
17163adant3 1008 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  (
a H Z )  =  Z )
186rneqi 5087 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ran  G  =  ran  ( 1st `  R
)
198, 18eqtri 2463 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  X  =  ran  ( 1st `  R
)
20 zerdivempx.5 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  U  =  (GId `  H )
217, 19, 20rngolidm 23933 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  ( U H B )  =  B )
22213adant2 1007 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( U H B )  =  B )
23 simp1 988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  ( a H Z ) )
24 simp2 989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  B )
25 simp3 990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( a H Z )  =  Z )
2623, 24, 253eqtr3d 2483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( A  e.  X  ->  ( ( U H B )  =  ( a H Z )  ->  B  =  Z )
) )
32313exp 1186 . . . . . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . 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 1181 . . . . . . . . . . . . 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 33 . . . . . . . . . . 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 1200 . . . . . . . . 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 1203 . . . . . . 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 30 . . . . . 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 1186 . . . 4  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  X  ->  ( R  e.  RingOps  -> 
( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) ) ) ) )
5150rexlimiv 2856 . . 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 1181 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2737   ran crn 4862   ` cfv 5439  (class class class)co 6112   1stc1st 6596   2ndc2nd 6597  GIdcgi 23696   RingOpscrngo 23884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fo 5445  df-fv 5447  df-riota 6073  df-ov 6115  df-1st 6598  df-2nd 6599  df-grpo 23700  df-gid 23701  df-ablo 23791  df-ass 23822  df-exid 23824  df-mgm 23828  df-sgr 23840  df-mndo 23847  df-rngo 23885
This theorem is referenced by:  isdrngo2  28790
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