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Theorem kerf1hrm 16953
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 5714 . . . . . . . . . . 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 5931 . . . . . . . . . 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 5808 . . . . . . . . 9  |-  ( F `
 x )  e. 
_V
109elsnc 4008 . . . . . . . 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 16950 . . . . . . . . . 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 1188 . . . . . . . 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 1218 . . . . . 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 3998 . . . . 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 3469 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( `' F " {  .0.  } )  C_  { N } )
25 rhmrcl1 16931 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
26 rnggrp 16772 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2712, 15grpidcl 15684 . . . . . . 7  |-  ( R  e.  Grp  ->  N  e.  A )
2825, 26, 273syl 20 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  N  e.  A )
29 rhmghm 16937 . . . . . . . 8  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
3015, 14ghmid 15871 . . . . . . . 8  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  N )  =  .0.  )
3129, 30syl 16 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  N )  =  .0.  )
32 fvex 5808 . . . . . . . 8  |-  ( F `
 N )  e. 
_V
3332elsnc 4008 . . . . . . 7  |-  ( ( F `  N )  e.  {  .0.  }  <->  ( F `  N )  =  .0.  )
3431, 33sylibr 212 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  N )  e.  {  .0.  } )
3512, 13rhmf 16938 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F : A
--> B )
36 ffn 5666 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
37 elpreima 5931 . . . . . . 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 913 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  N  e.  ( `' F " {  .0.  } ) )
4039snssd 4125 . . . 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 3480 . 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 1047 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  x  e.  A )
46 simpr2r 1048 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  y  e.  A )
47 simpr3 996 . . . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  ( `' F " {  .0.  } )  =  ( `' F " {  .0.  } )
49 eqid 2454 . . . . . . . . . . . 12  |-  ( -g `  R )  =  (
-g `  R )
5012, 14, 48, 49ghmeqker 15891 . . . . . . . . . . 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 1222 . . . . . . . . 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 994 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  ( `' F " {  .0.  } )  =  { N } )
5452, 53eleqtrd 2544 . . . . . . . 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 6224 . . . . . . . . 9  |-  ( x ( -g `  R
) y )  e. 
_V
5655elsnc 4008 . . . . . . . 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 15730 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  A  /\  y  e.  A )  ->  ( ( x (
-g `  R )
y )  =  N  <-> 
x  =  y ) )
6159, 45, 46, 60syl3anc 1219 . . . . . . 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 1210 . . . . 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 2912 . . 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 6079 . . 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 828 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 965    = wceq 1370    e. wcel 1758   A.wral 2798    C_ wss 3435   {csn 3984   `'ccnv 4946   "cima 4950    Fn wfn 5520   -->wf 5521   -1-1->wf1 5522   ` cfv 5525  (class class class)co 6199   Basecbs 14291   0gc0g 14496   Grpcgrp 15528   -gcsg 15531    GrpHom cghm 15862   Ringcrg 16767   RingHom crh 16926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-plusg 14369  df-0g 14498  df-mnd 15533  df-mhm 15582  df-grp 15663  df-minusg 15664  df-sbg 15665  df-ghm 15863  df-mgp 16713  df-ur 16725  df-rng 16769  df-rnghom 16928
This theorem is referenced by:  zrhf1ker  26548
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