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Theorem rngoridfz 21155
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 5908 . . . . . . . 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 21122 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
7 eqeq12 2328 . . . . . . . . . . . . . . . 16  |-  ( ( ( a H A )  =  U  /\  ( a H Z )  =  Z )  ->  ( ( a H A )  =  ( a H Z )  <->  U  =  Z
) )
87biimpd 198 . . . . . . . . . . . . . . 15  |-  ( ( ( a H A )  =  U  /\  ( a H Z )  =  Z )  ->  ( ( a H A )  =  ( a H Z )  ->  U  =  Z ) )
98ex 423 . . . . . . . . . . . . . 14  |-  ( ( a H A )  =  U  ->  (
( a H Z )  =  Z  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
106, 9syl5com 26 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
( a H A )  =  U  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
1110ex 423 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( a  e.  X  ->  ( (
a H A )  =  U  ->  (
( a H A )  =  ( a H Z )  ->  U  =  Z )
) ) )
1211com3l 75 . . . . . . . . . . 11  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( R  e.  RingOps  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) ) )
1312imp 418 . . . . . . . . . 10  |-  ( ( a  e.  X  /\  ( a H A )  =  U )  ->  ( R  e.  RingOps 
->  ( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
14133adant3 975 . . . . . . . . 9  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  ->  ( R  e.  RingOps  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
1514imp 418 . . . . . . . 8  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) )
161, 15syl5 28 . . . . . . 7  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( A  =  Z  ->  U  =  Z ) )
17 oveq2 5908 . . . . . . . 8  |-  ( U  =  Z  ->  ( A H U )  =  ( A H Z ) )
184rneqi 4942 . . . . . . . . . . . . . 14  |-  ran  G  =  ran  ( 1st `  R
)
193, 18eqtri 2336 . . . . . . . . . . . . 13  |-  X  =  ran  ( 1st `  R
)
20 zerdivemp.5 . . . . . . . . . . . . 13  |-  U  =  (GId `  H )
215, 19, 20rngoridm 21145 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
222, 3, 4, 5rngorz 21122 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  Z )
23 eqeq12 2328 . . . . . . . . . . . . 13  |-  ( ( ( A H U )  =  A  /\  ( A H Z )  =  Z )  -> 
( ( A H U )  =  ( A H Z )  <-> 
A  =  Z ) )
2423biimpd 198 . . . . . . . . . . . 12  |-  ( ( ( A H U )  =  A  /\  ( A H Z )  =  Z )  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) )
2521, 22, 24syl2anc 642 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H U )  =  ( A H Z )  ->  A  =  Z )
)
2625expcom 424 . . . . . . . . . 10  |-  ( A  e.  X  ->  ( R  e.  RingOps  ->  (
( A H U )  =  ( A H Z )  ->  A  =  Z )
) )
27263ad2ant3 978 . . . . . . . . 9  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  ->  ( R  e.  RingOps  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) ) )
2827imp 418 . . . . . . . 8  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) )
2917, 28syl5 28 . . . . . . 7  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( U  =  Z  ->  A  =  Z ) )
3016, 29impbid 183 . . . . . 6  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( A  =  Z  <-> 
U  =  Z ) )
31303exp1 1167 . . . . 5  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  X  ->  ( R  e.  RingOps  -> 
( A  =  Z  <-> 
U  =  Z ) ) ) ) )
3231rexlimiv 2695 . . . 4  |-  ( E. a  e.  X  ( a H A )  =  U  ->  ( A  e.  X  ->  ( R  e.  RingOps  ->  ( A  =  Z  <->  U  =  Z ) ) ) )
3332com13 74 . . 3  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( E. a  e.  X  (
a H A )  =  U  ->  ( A  =  Z  <->  U  =  Z ) ) ) )
34333imp 1145 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =  Z  <->  U  =  Z
) )
3534necon3bid 2514 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =/=  Z  <->  U  =/=  Z
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578   ran crn 4727   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163  GIdcgi 20907   RingOpscrngo 21095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fo 5298  df-fv 5300  df-ov 5903  df-1st 6164  df-2nd 6165  df-riota 6346  df-grpo 20911  df-gid 20912  df-ablo 21002  df-ass 21033  df-exid 21035  df-mgm 21039  df-sgr 21051  df-mndo 21058  df-rngo 21096
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