Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngokerinj Structured version   Unicode version

Theorem rngokerinj 26582
Description: A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
rngkerinj.1  |-  G  =  ( 1st `  R
)
rngkerinj.2  |-  X  =  ran  G
rngkerinj.3  |-  W  =  (GId `  G )
rngkerinj.4  |-  J  =  ( 1st `  S
)
rngkerinj.5  |-  Y  =  ran  J
rngkerinj.6  |-  Z  =  (GId `  J )
Assertion
Ref Expression
rngokerinj  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : X -1-1-> Y  <->  ( `' F " { Z } )  =  { W }
) )

Proof of Theorem rngokerinj
StepHypRef Expression
1 eqid 2435 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
21rngogrpo 21970 . . 3  |-  ( R  e.  RingOps  ->  ( 1st `  R
)  e.  GrpOp )
323ad2ant1 978 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( 1st `  R )  e.  GrpOp )
4 eqid 2435 . . . 4  |-  ( 1st `  S )  =  ( 1st `  S )
54rngogrpo 21970 . . 3  |-  ( S  e.  RingOps  ->  ( 1st `  S
)  e.  GrpOp )
653ad2ant2 979 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( 1st `  S )  e.  GrpOp )
71, 4rngogrphom 26578 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( ( 1st `  R
) GrpOpHom  ( 1st `  S
) ) )
8 rngkerinj.2 . . . 4  |-  X  =  ran  G
9 rngkerinj.1 . . . . 5  |-  G  =  ( 1st `  R
)
109rneqi 5088 . . . 4  |-  ran  G  =  ran  ( 1st `  R
)
118, 10eqtri 2455 . . 3  |-  X  =  ran  ( 1st `  R
)
12 rngkerinj.3 . . . 4  |-  W  =  (GId `  G )
139fveq2i 5723 . . . 4  |-  (GId `  G )  =  (GId
`  ( 1st `  R
) )
1412, 13eqtri 2455 . . 3  |-  W  =  (GId `  ( 1st `  R ) )
15 rngkerinj.5 . . . 4  |-  Y  =  ran  J
16 rngkerinj.4 . . . . 5  |-  J  =  ( 1st `  S
)
1716rneqi 5088 . . . 4  |-  ran  J  =  ran  ( 1st `  S
)
1815, 17eqtri 2455 . . 3  |-  Y  =  ran  ( 1st `  S
)
19 rngkerinj.6 . . . 4  |-  Z  =  (GId `  J )
2016fveq2i 5723 . . . 4  |-  (GId `  J )  =  (GId
`  ( 1st `  S
) )
2119, 20eqtri 2455 . . 3  |-  Z  =  (GId `  ( 1st `  S ) )
2211, 14, 18, 21grpokerinj 26551 . 2  |-  ( ( ( 1st `  R
)  e.  GrpOp  /\  ( 1st `  S )  e. 
GrpOp  /\  F  e.  ( ( 1st `  R
) GrpOpHom  ( 1st `  S
) ) )  -> 
( F : X -1-1-> Y  <-> 
( `' F " { Z } )  =  { W } ) )
233, 6, 7, 22syl3anc 1184 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : X -1-1-> Y  <->  ( `' F " { Z } )  =  { W }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   {csn 3806   `'ccnv 4869   ran crn 4871   "cima 4873   -1-1->wf1 5443   ` cfv 5446  (class class class)co 6073   1stc1st 6339   GrpOpcgr 21766  GIdcgi 21767   GrpOpHom cghom 21937   RingOpscrngo 21955    RngHom crnghom 26567
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-map 7012  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-ghom 21938  df-rngo 21956  df-rngohom 26570
  Copyright terms: Public domain W3C validator