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Theorem rngokerinj 26709
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 2296 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
21rngogrpo 21073 . . 3  |-  ( R  e.  RingOps  ->  ( 1st `  R
)  e.  GrpOp )
323ad2ant1 976 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( 1st `  R )  e.  GrpOp )
4 eqid 2296 . . . 4  |-  ( 1st `  S )  =  ( 1st `  S )
54rngogrpo 21073 . . 3  |-  ( S  e.  RingOps  ->  ( 1st `  S
)  e.  GrpOp )
653ad2ant2 977 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( 1st `  S )  e.  GrpOp )
71, 4rngogrphom 26705 . 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 4921 . . . 4  |-  ran  G  =  ran  ( 1st `  R
)
118, 10eqtri 2316 . . 3  |-  X  =  ran  ( 1st `  R
)
12 rngkerinj.3 . . . 4  |-  W  =  (GId `  G )
139fveq2i 5544 . . . 4  |-  (GId `  G )  =  (GId
`  ( 1st `  R
) )
1412, 13eqtri 2316 . . 3  |-  W  =  (GId `  ( 1st `  R ) )
15 rngkerinj.5 . . . 4  |-  Y  =  ran  J
16 rngkerinj.4 . . . . 5  |-  J  =  ( 1st `  S
)
1716rneqi 4921 . . . 4  |-  ran  J  =  ran  ( 1st `  S
)
1815, 17eqtri 2316 . . 3  |-  Y  =  ran  ( 1st `  S
)
19 rngkerinj.6 . . . 4  |-  Z  =  (GId `  J )
2016fveq2i 5544 . . . 4  |-  (GId `  J )  =  (GId
`  ( 1st `  S
) )
2119, 20eqtri 2316 . . 3  |-  Z  =  (GId `  ( 1st `  S ) )
2211, 14, 18, 21grpokerinj 26678 . 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 1182 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 176    /\ w3a 934    = wceq 1632    e. wcel 1696   {csn 3653   `'ccnv 4704   ran crn 4706   "cima 4708   -1-1->wf1 5268   ` cfv 5271  (class class class)co 5874   1stc1st 6136   GrpOpcgr 20869  GIdcgi 20870   GrpOpHom cghom 21040   RingOpscrngo 21058    RngHom crnghom 26694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-map 6790  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-ghom 21041  df-rngo 21059  df-rngohom 26697
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