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Theorem rngoridfz 25413
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 6289 . . . . . . . 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 25380 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
7 eqeq12 2462 . . . . . . . . . . . . . . . 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 1017 . . . . . . . . 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 6289 . . . . . . . 8  |-  ( U  =  Z  ->  ( A H U )  =  ( A H Z ) )
184rneqi 5219 . . . . . . . . . . . . . 14  |-  ran  G  =  ran  ( 1st `  R
)
193, 18eqtri 2472 . . . . . . . . . . . . 13  |-  X  =  ran  ( 1st `  R
)
20 zerdivemp.5 . . . . . . . . . . . . 13  |-  U  =  (GId `  H )
215, 19, 20rngoridm 25403 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
222, 3, 4, 5rngorz 25380 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  Z )
23 eqeq12 2462 . . . . . . . . . . . . 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 1020 . . . . . . . . 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 1213 . . . . 5  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  X  ->  ( R  e.  RingOps  -> 
( A  =  Z  <-> 
U  =  Z ) ) ) ) )
3231rexlimiv 2929 . . . 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 1191 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =  Z  <->  U  =  Z
) )
3534necon3bid 2701 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   ran crn 4990   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784  GIdcgi 25165   RingOpscrngo 25353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-riota 6242  df-ov 6284  df-1st 6785  df-2nd 6786  df-grpo 25169  df-gid 25170  df-ablo 25260  df-ass 25291  df-exid 25293  df-mgmOLD 25297  df-sgrOLD 25309  df-mndo 25316  df-rngo 25354
This theorem is referenced by: (None)
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