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Theorem rngoridfz 22023
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 6089 . . . . . . . 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 21990 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
7 eqeq12 2448 . . . . . . . . . . . . . . . 16  |-  ( ( ( a H A )  =  U  /\  ( a H Z )  =  Z )  ->  ( ( a H A )  =  ( a H Z )  <->  U  =  Z
) )
87biimpd 199 . . . . . . . . . . . . . . 15  |-  ( ( ( a H A )  =  U  /\  ( a H Z )  =  Z )  ->  ( ( a H A )  =  ( a H Z )  ->  U  =  Z ) )
98ex 424 . . . . . . . . . . . . . 14  |-  ( ( a H A )  =  U  ->  (
( a H Z )  =  Z  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
106, 9syl5com 28 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
( a H A )  =  U  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
1110ex 424 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( a  e.  X  ->  ( (
a H A )  =  U  ->  (
( a H A )  =  ( a H Z )  ->  U  =  Z )
) ) )
1211com3l 77 . . . . . . . . . . 11  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( R  e.  RingOps  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) ) )
1312imp 419 . . . . . . . . . 10  |-  ( ( a  e.  X  /\  ( a H A )  =  U )  ->  ( R  e.  RingOps 
->  ( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
14133adant3 977 . . . . . . . . 9  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  ->  ( R  e.  RingOps  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) ) )
1514imp 419 . . . . . . . 8  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( ( a H A )  =  ( a H Z )  ->  U  =  Z ) )
161, 15syl5 30 . . . . . . 7  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( A  =  Z  ->  U  =  Z ) )
17 oveq2 6089 . . . . . . . 8  |-  ( U  =  Z  ->  ( A H U )  =  ( A H Z ) )
184rneqi 5096 . . . . . . . . . . . . . 14  |-  ran  G  =  ran  ( 1st `  R
)
193, 18eqtri 2456 . . . . . . . . . . . . 13  |-  X  =  ran  ( 1st `  R
)
20 zerdivemp.5 . . . . . . . . . . . . 13  |-  U  =  (GId `  H )
215, 19, 20rngoridm 22013 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H U )  =  A )
222, 3, 4, 5rngorz 21990 . . . . . . . . . . . 12  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( A H Z )  =  Z )
23 eqeq12 2448 . . . . . . . . . . . . 13  |-  ( ( ( A H U )  =  A  /\  ( A H Z )  =  Z )  -> 
( ( A H U )  =  ( A H Z )  <-> 
A  =  Z ) )
2423biimpd 199 . . . . . . . . . . . 12  |-  ( ( ( A H U )  =  A  /\  ( A H Z )  =  Z )  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) )
2521, 22, 24syl2anc 643 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  (
( A H U )  =  ( A H Z )  ->  A  =  Z )
)
2625expcom 425 . . . . . . . . . 10  |-  ( A  e.  X  ->  ( R  e.  RingOps  ->  (
( A H U )  =  ( A H Z )  ->  A  =  Z )
) )
27263ad2ant3 980 . . . . . . . . 9  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  ->  ( R  e.  RingOps  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) ) )
2827imp 419 . . . . . . . 8  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( ( A H U )  =  ( A H Z )  ->  A  =  Z ) )
2917, 28syl5 30 . . . . . . 7  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( U  =  Z  ->  A  =  Z ) )
3016, 29impbid 184 . . . . . 6  |-  ( ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  X )  /\  R  e.  RingOps )  -> 
( A  =  Z  <-> 
U  =  Z ) )
31303exp1 1169 . . . . 5  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  X  ->  ( R  e.  RingOps  -> 
( A  =  Z  <-> 
U  =  Z ) ) ) ) )
3231rexlimiv 2824 . . . 4  |-  ( E. a  e.  X  ( a H A )  =  U  ->  ( A  e.  X  ->  ( R  e.  RingOps  ->  ( A  =  Z  <->  U  =  Z ) ) ) )
3332com13 76 . . 3  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( E. a  e.  X  (
a H A )  =  U  ->  ( A  =  Z  <->  U  =  Z ) ) ) )
34333imp 1147 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =  Z  <->  U  =  Z
) )
3534necon3bid 2636 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   ran crn 4879   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348  GIdcgi 21775   RingOpscrngo 21963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ablo 21870  df-ass 21901  df-exid 21903  df-mgm 21907  df-sgr 21919  df-mndo 21926  df-rngo 21964
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