MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngoridfz Structured version   Unicode version

Theorem rngoridfz 25101
Description: In a unitary ring a left invertible element is different from zero iff  1  =/=  0. (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
rngoridfz  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =/=  Z  <->  U  =/=  Z
) )
Distinct variable groups:    A, a    R, a    U, a    X, a    Z, a
Allowed substitution hints:    G( a)    H( a)

Proof of Theorem rngoridfz
StepHypRef Expression
1 oveq2 6285 . . . . . . . 8  |-  ( A  =  Z  ->  (
a H A )  =  ( a H Z ) )
2 zerdivemp.3 . . . . . . . . . . . . . . 15  |-  Z  =  (GId `  G )
3 zerdivemp.4 . . . . . . . . . . . . . . 15  |-  X  =  ran  G
4 zerdivemp.1 . . . . . . . . . . . . . . 15  |-  G  =  ( 1st `  R
)
5 zerdivemp.2 . . . . . . . . . . . . . . 15  |-  H  =  ( 2nd `  R
)
62, 3, 4, 5rngorz 25068 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
7 eqeq12 2481 . . . . . . . . . . . . . . . 16  |-  ( ( ( a H A )  =  U  /\  ( a H Z )  =  Z )  ->  ( ( a H A )  =  ( a H Z )  <->  U  =  Z
) )
87biimpd 207 . . . . . . . . . . . . . . 15  |-  ( ( ( a H A )  =  U  /\  ( a H Z )  =  Z )  ->  ( ( a H A )  =  ( a H Z )  ->  U  =  Z ) )
98ex 434 . . . . . . . . . . . . . 14  |-  ( ( a H A )  =  U  ->  (
( a H Z )  =  Z  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
106, 9syl5com 30 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
( a H A )  =  U  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
1110ex 434 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( a  e.  X  ->  ( (
a H A )  =  U  ->  (
( a H A )  =  ( a H Z )  ->  U  =  Z )
) ) )
1211com3l 81 . . . . . . . . . . 11  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( R  e.  RingOps  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) ) )
1312imp 429 . . . . . . . . . 10  |-  ( ( a  e.  X  /\  ( a H A )  =  U )  ->  ( R  e.  RingOps 
->  ( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
14133adant3 1011 . . . . . . . . 9  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  ->  ( R  e.  RingOps  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
1514imp 429 . . . . . . . 8  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) )
161, 15syl5 32 . . . . . . 7  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( A  =  Z  ->  U  =  Z ) )
17 oveq2 6285 . . . . . . . 8  |-  ( U  =  Z  ->  ( A H U )  =  ( A H Z ) )
184rneqi 5222 . . . . . . . . . . . . . 14  |-  ran  G  =  ran  ( 1st `  R
)
193, 18eqtri 2491 . . . . . . . . . . . . 13  |-  X  =  ran  ( 1st `  R
)
20 zerdivemp.5 . . . . . . . . . . . . 13  |-  U  =  (GId `  H )
215, 19, 20rngoridm 25091 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
222, 3, 4, 5rngorz 25068 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  Z )
23 eqeq12 2481 . . . . . . . . . . . . 13  |-  ( ( ( A H U )  =  A  /\  ( A H Z )  =  Z )  -> 
( ( A H U )  =  ( A H Z )  <-> 
A  =  Z ) )
2423biimpd 207 . . . . . . . . . . . 12  |-  ( ( ( A H U )  =  A  /\  ( A H Z )  =  Z )  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) )
2521, 22, 24syl2anc 661 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H U )  =  ( A H Z )  ->  A  =  Z )
)
2625expcom 435 . . . . . . . . . 10  |-  ( A  e.  X  ->  ( R  e.  RingOps  ->  (
( A H U )  =  ( A H Z )  ->  A  =  Z )
) )
27263ad2ant3 1014 . . . . . . . . 9  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  ->  ( R  e.  RingOps  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) ) )
2827imp 429 . . . . . . . 8  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) )
2917, 28syl5 32 . . . . . . 7  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( U  =  Z  ->  A  =  Z ) )
3016, 29impbid 191 . . . . . 6  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( A  =  Z  <-> 
U  =  Z ) )
31303exp1 1207 . . . . 5  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  X  ->  ( R  e.  RingOps  -> 
( A  =  Z  <-> 
U  =  Z ) ) ) ) )
3231rexlimiv 2944 . . . 4  |-  ( E. a  e.  X  ( a H A )  =  U  ->  ( A  e.  X  ->  ( R  e.  RingOps  ->  ( A  =  Z  <->  U  =  Z ) ) ) )
3332com13 80 . . 3  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( E. a  e.  X  (
a H A )  =  U  ->  ( A  =  Z  <->  U  =  Z ) ) ) )
34333imp 1185 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =  Z  <->  U  =  Z
) )
3534necon3bid 2720 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =/=  Z  <->  U  =/=  Z
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810   ran crn 4995   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775  GIdcgi 24853   RingOpscrngo 25041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-riota 6238  df-ov 6280  df-1st 6776  df-2nd 6777  df-grpo 24857  df-gid 24858  df-ablo 24948  df-ass 24979  df-exid 24981  df-mgm 24985  df-sgr 24997  df-mndo 25004  df-rngo 25042
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator