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Theorem zerdivemp1 25301
Description: In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
Hypotheses
Ref Expression
zerdivemp.1  |-  G  =  ( 1st `  R
)
zerdivemp.2  |-  H  =  ( 2nd `  R
)
zerdivemp.3  |-  Z  =  (GId `  G )
zerdivemp.4  |-  X  =  ran  G
zerdivemp.5  |-  U  =  (GId `  H )
Assertion
Ref Expression
zerdivemp1  |-  ( ( R  e.  RingOps  /\  A  e.  ( X  \  Z
)  /\  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 zerdivemp1
StepHypRef Expression
1 oveq2 6285 . . . . . . 7  |-  ( ( A H B )  =  Z  ->  (
a H ( A H B ) )  =  ( a H Z ) )
2 simpl1 998 . . . . . . . . . 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  \  Z ) ) )  ->  R  e.  RingOps )
3 simpr1 1001 . . . . . . . . . 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  \  Z ) ) )  ->  a  e.  X
)
4 eldifi 3608 . . . . . . . . . . . 12  |-  ( A  e.  ( X  \  Z )  ->  A  e.  X )
543ad2ant3 1018 . . . . . . . . . . 11  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  ->  A  e.  X )
65adantl 466 . . . . . . . . . 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  \  Z ) ) )  ->  A  e.  X
)
7 simpl3 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  \  Z ) ) )  ->  B  e.  X
)
8 zerdivemp.1 . . . . . . . . . . 11  |-  G  =  ( 1st `  R
)
9 zerdivemp.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
10 zerdivemp.4 . . . . . . . . . . 11  |-  X  =  ran  G
118, 9, 10rngoass 25254 . . . . . . . . . 10  |-  ( ( R  e.  RingOps  /\  (
a  e.  X  /\  A  e.  X  /\  B  e.  X )
)  ->  ( (
a H A ) H B )  =  ( a H ( A H B ) ) )
122, 3, 6, 7, 11syl13anc 1229 . . . . . . . . 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  \  Z ) ) )  ->  ( ( a H A ) H B )  =  ( a H ( A H B ) ) )
13 eqtr 2467 . . . . . . . . . . . . 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 ) )
1413ex 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 ) ) )
15 oveq1 6284 . . . . . . . . . . . . . . . 16  |-  ( ( a H A )  =  U  ->  (
( a H A ) H B )  =  ( U H B ) )
16 eqtr 2467 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( U H B )  =  ( ( a H A ) H B )  /\  ( ( a H A ) H B )  =  ( a H Z ) )  ->  ( U H B )  =  ( a H Z ) )
17 zerdivemp.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  Z  =  (GId `  G )
1817, 10, 8, 9rngorz 25269 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
19183adant3 1015 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  (
a H Z )  =  Z )
208rneqi 5215 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ran  G  =  ran  ( 1st `  R
)
2110, 20eqtri 2470 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  X  =  ran  ( 1st `  R
)
22 zerdivemp.5 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  U  =  (GId `  H )
239, 21, 22rngolidm 25291 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  ( U H B )  =  B )
24233adant2 1014 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( U H B )  =  B )
25 simp1 995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  ( a H Z ) )
26 simp2 996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  B )
27 simp3 997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( a H Z )  =  Z )
2825, 26, 273eqtr3d 2490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  ->  B  =  Z )
2928a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( A  e.  ( X  \  Z )  ->  B  =  Z ) )
30293exp 1194 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( U H B )  =  ( a H Z )  ->  (
( U H B )  =  B  -> 
( ( a H Z )  =  Z  ->  ( A  e.  ( X  \  Z
)  ->  B  =  Z ) ) ) )
3130com14 88 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( A  e.  ( X  \  Z )  ->  (
( U H B )  =  B  -> 
( ( a H Z )  =  Z  ->  ( ( U H B )  =  ( a H Z )  ->  B  =  Z ) ) ) )
3231com13 80 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( a H Z )  =  Z  ->  (
( U H B )  =  B  -> 
( A  e.  ( X  \  Z )  ->  ( ( U H B )  =  ( a H Z )  ->  B  =  Z ) ) ) )
3319, 24, 32sylc 60 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( A  e.  ( X  \  Z )  ->  (
( U H B )  =  ( a H Z )  ->  B  =  Z )
) )
34333exp 1194 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( R  e.  RingOps  ->  ( a  e.  X  ->  ( B  e.  X  ->  ( A  e.  ( X  \  Z )  ->  (
( U H B )  =  ( a H Z )  ->  B  =  Z )
) ) ) )
3534com15 93 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U H B )  =  ( a H Z )  ->  (
a  e.  X  -> 
( B  e.  X  ->  ( A  e.  ( X  \  Z )  ->  ( R  e.  RingOps 
->  B  =  Z
) ) ) ) )
3635com24 87 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U H B )  =  ( a H Z )  ->  ( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( a  e.  X  -> 
( R  e.  RingOps  ->  B  =  Z )
) ) ) )
3716, 36syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( U H B )  =  ( ( a H A ) H B )  /\  ( ( a H A ) H B )  =  ( a H Z ) )  ->  ( A  e.  ( X  \  Z
)  ->  ( B  e.  X  ->  ( a  e.  X  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) )
3837ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( ( U H B )  =  ( ( a H A ) H B )  ->  (
( ( a H A ) H B )  =  ( a H Z )  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( a  e.  X  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
3938eqcoms 2453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a H A ) H B )  =  ( U H B )  ->  (
( ( a H A ) H B )  =  ( a H Z )  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( a  e.  X  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
4039com25 91 . . . . . . . . . . . . . . . 16  |-  ( ( ( a H A ) H B )  =  ( U H B )  ->  (
a  e.  X  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( (
( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
4115, 40syl 16 . . . . . . . . . . . . . . 15  |-  ( ( a H A )  =  U  ->  (
a  e.  X  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( (
( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
4241com12 31 . . . . . . . . . . . . . 14  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( (
( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
43423imp 1189 . . . . . . . . . . . . 13  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  -> 
( B  e.  X  ->  ( ( ( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps 
->  B  =  Z
) ) ) )
4443com13 80 . . . . . . . . . . . 12  |-  ( ( ( a H A ) H B )  =  ( a H Z )  ->  ( B  e.  X  ->  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  -> 
( R  e.  RingOps  ->  B  =  Z )
) ) )
4514, 44syl6 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  \  Z
) )  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) )
4645com15 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  \  Z ) )  -> 
( ( ( a H A ) H B )  =  ( a H ( A H B ) )  ->  B  =  Z ) ) ) ) )
47463imp1 1208 . . . . . . . . 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  \  Z ) ) )  ->  ( ( ( a H A ) H B )  =  ( a H ( A H B ) )  ->  B  =  Z ) )
4812, 47mpd 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  \  Z ) ) )  ->  B  =  Z )
49483exp1 1211 . . . . . . 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  \  Z ) )  ->  B  =  Z )
) ) )
501, 49syl5com 30 . . . . . 6  |-  ( ( A H B )  =  Z  ->  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  ->  B  =  Z )
) ) )
5150com14 88 . . . . 5  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  -> 
( R  e.  RingOps  -> 
( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) ) )
52513exp 1194 . . . 4  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  ( X  \  Z )  ->  ( R  e.  RingOps 
->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) ) ) ) )
5352rexlimiv 2927 . . 3  |-  ( E. a  e.  X  ( a H A )  =  U  ->  ( A  e.  ( X  \  Z )  ->  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z )
) ) ) )
5453com13 80 . 2  |-  ( R  e.  RingOps  ->  ( A  e.  ( X  \  Z
)  ->  ( E. a  e.  X  (
a H A )  =  U  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z )
) ) ) )
55543imp 1189 1  |-  ( ( R  e.  RingOps  /\  A  e.  ( X  \  Z
)  /\  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 972    = wceq 1381    e. wcel 1802   E.wrex 2792    \ cdif 3455   ran crn 4986   ` cfv 5574  (class class class)co 6277   1stc1st 6779   2ndc2nd 6780  GIdcgi 25054   RingOpscrngo 25242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fo 5580  df-fv 5582  df-riota 6238  df-ov 6280  df-1st 6781  df-2nd 6782  df-grpo 25058  df-gid 25059  df-ablo 25149  df-ass 25180  df-exid 25182  df-mgmOLD 25186  df-sgrOLD 25198  df-mndo 25205  df-rngo 25243
This theorem is referenced by: (None)
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