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Theorem kerf1hrm 17263
Description: A ring homomorphism  F is injective if and only if its kernel is the singleton  { N }. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.)
Hypotheses
Ref Expression
kerf1hrm.a  |-  A  =  ( Base `  R
)
kerf1hrm.b  |-  B  =  ( Base `  S
)
kerf1hrm.n  |-  N  =  ( 0g `  R
)
kerf1hrm.0  |-  .0.  =  ( 0g `  S )
Assertion
Ref Expression
kerf1hrm  |-  ( F  e.  ( R RingHom  S
)  ->  ( F : A -1-1-> B  <->  ( `' F " {  .0.  } )  =  { N }
) )

Proof of Theorem kerf1hrm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . . . 7  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B ) )
2 f1fn 5788 . . . . . . . . . . 11  |-  ( F : A -1-1-> B  ->  F  Fn  A )
32adantl 466 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  ->  F  Fn  A )
4 elpreima 6008 . . . . . . . . . 10  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " {  .0.  } )  <->  ( x  e.  A  /\  ( F `
 x )  e. 
{  .0.  } ) ) )
53, 4syl 16 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( x  e.  ( `' F " {  .0.  } )  <->  ( x  e.  A  /\  ( F `
 x )  e. 
{  .0.  } ) ) )
65biimpa 484 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  (
x  e.  A  /\  ( F `  x )  e.  {  .0.  }
) )
76simpld 459 . . . . . . 7  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  x  e.  A )
86simprd 463 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  ( F `  x )  e.  {  .0.  } )
9 fvex 5882 . . . . . . . . 9  |-  ( F `
 x )  e. 
_V
109elsnc 4057 . . . . . . . 8  |-  ( ( F `  x )  e.  {  .0.  }  <->  ( F `  x )  =  .0.  )
118, 10sylib 196 . . . . . . 7  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  ( F `  x )  =  .0.  )
12 kerf1hrm.a . . . . . . . . . . 11  |-  A  =  ( Base `  R
)
13 kerf1hrm.b . . . . . . . . . . 11  |-  B  =  ( Base `  S
)
14 kerf1hrm.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  S )
15 kerf1hrm.n . . . . . . . . . . 11  |-  N  =  ( 0g `  R
)
1612, 13, 14, 15f1rhm0to0 17260 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B  /\  x  e.  A )  ->  (
( F `  x
)  =  .0.  <->  x  =  N ) )
1716biimpd 207 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B  /\  x  e.  A )  ->  (
( F `  x
)  =  .0.  ->  x  =  N ) )
18173expa 1196 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  A )  ->  ( ( F `  x )  =  .0. 
->  x  =  N
) )
1918imp 429 . . . . . . 7  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B
)  /\  x  e.  A )  /\  ( F `  x )  =  .0.  )  ->  x  =  N )
201, 7, 11, 19syl21anc 1227 . . . . . 6  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  x  =  N )
2120ex 434 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( x  e.  ( `' F " {  .0.  } )  ->  x  =  N ) )
22 elsn 4047 . . . . 5  |-  ( x  e.  { N }  <->  x  =  N )
2321, 22syl6ibr 227 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( x  e.  ( `' F " {  .0.  } )  ->  x  e.  { N } ) )
2423ssrdv 3515 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( `' F " {  .0.  } )  C_  { N } )
25 rhmrcl1 17240 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
26 ringgrp 17075 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2712, 15grpidcl 15950 . . . . . . 7  |-  ( R  e.  Grp  ->  N  e.  A )
2825, 26, 273syl 20 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  N  e.  A )
29 rhmghm 17246 . . . . . . . 8  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
3015, 14ghmid 16145 . . . . . . . 8  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  N )  =  .0.  )
3129, 30syl 16 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  N )  =  .0.  )
32 fvex 5882 . . . . . . . 8  |-  ( F `
 N )  e. 
_V
3332elsnc 4057 . . . . . . 7  |-  ( ( F `  N )  e.  {  .0.  }  <->  ( F `  N )  =  .0.  )
3431, 33sylibr 212 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  N )  e.  {  .0.  } )
3512, 13rhmf 17247 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F : A
--> B )
36 ffn 5737 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
37 elpreima 6008 . . . . . . 7  |-  ( F  Fn  A  ->  ( N  e.  ( `' F " {  .0.  }
)  <->  ( N  e.  A  /\  ( F `
 N )  e. 
{  .0.  } ) ) )
3835, 36, 373syl 20 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  ( N  e.  ( `' F " {  .0.  } )  <->  ( N  e.  A  /\  ( F `  N )  e.  {  .0.  } ) ) )
3928, 34, 38mpbir2and 920 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  N  e.  ( `' F " {  .0.  } ) )
4039snssd 4178 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  { N }  C_  ( `' F " {  .0.  } ) )
4140adantr 465 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  ->  { N }  C_  ( `' F " {  .0.  } ) )
4224, 41eqssd 3526 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( `' F " {  .0.  } )  =  { N } )
4335adantr 465 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  ->  F : A --> B )
4429adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  F  e.  ( R  GrpHom  S ) )
45 simpr2l 1055 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  x  e.  A )
46 simpr2r 1056 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  y  e.  A )
47 simpr3 1004 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  ( F `  x )  =  ( F `  y ) )
48 eqid 2467 . . . . . . . . . . . 12  |-  ( `' F " {  .0.  } )  =  ( `' F " {  .0.  } )
49 eqid 2467 . . . . . . . . . . . 12  |-  ( -g `  R )  =  (
-g `  R )
5012, 14, 48, 49ghmeqker 16165 . . . . . . . . . . 11  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  x  e.  A  /\  y  e.  A )  ->  (
( F `  x
)  =  ( F `
 y )  <->  ( x
( -g `  R ) y )  e.  ( `' F " {  .0.  } ) ) )
5150biimpa 484 . . . . . . . . . 10  |-  ( ( ( F  e.  ( R  GrpHom  S )  /\  x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) )  ->  ( x (
-g `  R )
y )  e.  ( `' F " {  .0.  } ) )
5244, 45, 46, 47, 51syl31anc 1231 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  (
x ( -g `  R
) y )  e.  ( `' F " {  .0.  } ) )
53 simpr1 1002 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  ( `' F " {  .0.  } )  =  { N } )
5452, 53eleqtrd 2557 . . . . . . . 8  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  (
x ( -g `  R
) y )  e. 
{ N } )
55 ovex 6320 . . . . . . . . 9  |-  ( x ( -g `  R
) y )  e. 
_V
5655elsnc 4057 . . . . . . . 8  |-  ( ( x ( -g `  R
) y )  e. 
{ N }  <->  ( x
( -g `  R ) y )  =  N )
5754, 56sylib 196 . . . . . . 7  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  (
x ( -g `  R
) y )  =  N )
5825adantr 465 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  R  e.  Ring )
5958, 26syl 16 . . . . . . . 8  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  R  e.  Grp )
6012, 15, 49grpsubeq0 15996 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  A  /\  y  e.  A )  ->  ( ( x (
-g `  R )
y )  =  N  <-> 
x  =  y ) )
6159, 45, 46, 60syl3anc 1228 . . . . . . 7  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  (
( x ( -g `  R ) y )  =  N  <->  x  =  y ) )
6257, 61mpbid 210 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  x  =  y )
63623anassrs 1218 . . . . 5  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  /\  ( x  e.  A  /\  y  e.  A ) )  /\  ( F `  x )  =  ( F `  y ) )  ->  x  =  y )
6463ex 434 . . . 4  |-  ( ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
6564ralrimivva 2888 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  ->  A. x  e.  A  A. y  e.  A  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
66 dff13 6165 . . 3  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
6743, 65, 66sylanbrc 664 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  ->  F : A -1-1-> B )
6842, 67impbida 830 1  |-  ( F  e.  ( R RingHom  S
)  ->  ( F : A -1-1-> B  <->  ( `' F " {  .0.  } )  =  { N }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   {csn 4033   `'ccnv 5004   "cima 5008    Fn wfn 5589   -->wf 5590   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6295   Basecbs 14507   0gc0g 14712   Grpcgrp 15925   -gcsg 15927    GrpHom cghm 16136   Ringcrg 17070   RingHom crh 17233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-grp 15929  df-minusg 15930  df-sbg 15931  df-ghm 16137  df-mgp 17014  df-ur 17026  df-ring 17072  df-rnghom 17236
This theorem is referenced by:  zrhf1ker  27772
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