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Theorem kerf1hrm 17971
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 459 . . . . . . 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 5780 . . . . . . . . . . 11  |-  ( F : A -1-1-> B  ->  F  Fn  A )
32adantl 468 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  ->  F  Fn  A )
4 elpreima 6002 . . . . . . . . . 10  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " {  .0.  } )  <->  ( x  e.  A  /\  ( F `
 x )  e. 
{  .0.  } ) ) )
53, 4syl 17 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( x  e.  ( `' F " {  .0.  } )  <->  ( x  e.  A  /\  ( F `
 x )  e. 
{  .0.  } ) ) )
65biimpa 487 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  (
x  e.  A  /\  ( F `  x )  e.  {  .0.  }
) )
76simpld 461 . . . . . . 7  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  x  e.  A )
86simprd 465 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  ( F `  x )  e.  {  .0.  } )
9 fvex 5875 . . . . . . . . 9  |-  ( F `
 x )  e. 
_V
109elsnc 3992 . . . . . . . 8  |-  ( ( F `  x )  e.  {  .0.  }  <->  ( F `  x )  =  .0.  )
118, 10sylib 200 . . . . . . 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 17968 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B  /\  x  e.  A )  ->  (
( F `  x
)  =  .0.  <->  x  =  N ) )
1716biimpd 211 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B  /\  x  e.  A )  ->  (
( F `  x
)  =  .0.  ->  x  =  N ) )
18173expa 1208 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  A )  ->  ( ( F `  x )  =  .0. 
->  x  =  N
) )
1918imp 431 . . . . . . 7  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B
)  /\  x  e.  A )  /\  ( F `  x )  =  .0.  )  ->  x  =  N )
201, 7, 11, 19syl21anc 1267 . . . . . 6  |-  ( ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  /\  x  e.  ( `' F " {  .0.  }
) )  ->  x  =  N )
2120ex 436 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( x  e.  ( `' F " {  .0.  } )  ->  x  =  N ) )
22 elsn 3982 . . . . 5  |-  ( x  e.  { N }  <->  x  =  N )
2321, 22syl6ibr 231 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( x  e.  ( `' F " {  .0.  } )  ->  x  e.  { N } ) )
2423ssrdv 3438 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( `' F " {  .0.  } )  C_  { N } )
25 rhmrcl1 17947 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
26 ringgrp 17785 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2712, 15grpidcl 16694 . . . . . . 7  |-  ( R  e.  Grp  ->  N  e.  A )
2825, 26, 273syl 18 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  N  e.  A )
29 rhmghm 17953 . . . . . . . 8  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
3015, 14ghmid 16889 . . . . . . . 8  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  N )  =  .0.  )
3129, 30syl 17 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  N )  =  .0.  )
32 fvex 5875 . . . . . . . 8  |-  ( F `
 N )  e. 
_V
3332elsnc 3992 . . . . . . 7  |-  ( ( F `  N )  e.  {  .0.  }  <->  ( F `  N )  =  .0.  )
3431, 33sylibr 216 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  N )  e.  {  .0.  } )
3512, 13rhmf 17954 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F : A
--> B )
36 ffn 5728 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
37 elpreima 6002 . . . . . . 7  |-  ( F  Fn  A  ->  ( N  e.  ( `' F " {  .0.  }
)  <->  ( N  e.  A  /\  ( F `
 N )  e. 
{  .0.  } ) ) )
3835, 36, 373syl 18 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  ( N  e.  ( `' F " {  .0.  } )  <->  ( N  e.  A  /\  ( F `  N )  e.  {  .0.  } ) ) )
3928, 34, 38mpbir2and 933 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  N  e.  ( `' F " {  .0.  } ) )
4039snssd 4117 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  { N }  C_  ( `' F " {  .0.  } ) )
4140adantr 467 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  ->  { N }  C_  ( `' F " {  .0.  } ) )
4224, 41eqssd 3449 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  F : A -1-1-> B )  -> 
( `' F " {  .0.  } )  =  { N } )
4335adantr 467 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  ->  F : A --> B )
4429adantr 467 . . . . . . . . . 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 1067 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  x  e.  A )
46 simpr2r 1068 . . . . . . . . . 10  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  y  e.  A )
47 simpr3 1016 . . . . . . . . . 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 2451 . . . . . . . . . . . 12  |-  ( `' F " {  .0.  } )  =  ( `' F " {  .0.  } )
49 eqid 2451 . . . . . . . . . . . 12  |-  ( -g `  R )  =  (
-g `  R )
5012, 14, 48, 49ghmeqker 16909 . . . . . . . . . . 11  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  x  e.  A  /\  y  e.  A )  ->  (
( F `  x
)  =  ( F `
 y )  <->  ( x
( -g `  R ) y )  e.  ( `' F " {  .0.  } ) ) )
5150biimpa 487 . . . . . . . . . 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 1271 . . . . . . . . 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 1014 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  ( `' F " {  .0.  } )  =  { N } )
5452, 53eleqtrd 2531 . . . . . . . 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 6318 . . . . . . . . 9  |-  ( x ( -g `  R
) y )  e. 
_V
5655elsnc 3992 . . . . . . . 8  |-  ( ( x ( -g `  R
) y )  e. 
{ N }  <->  ( x
( -g `  R ) y )  =  N )
5754, 56sylib 200 . . . . . . 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 467 . . . . . . . . 9  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  R  e.  Ring )
5958, 26syl 17 . . . . . . . 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 16740 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  x  e.  A  /\  y  e.  A )  ->  ( ( x (
-g `  R )
y )  =  N  <-> 
x  =  y ) )
6159, 45, 46, 60syl3anc 1268 . . . . . . 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 214 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( `' F " {  .0.  } )  =  { N }  /\  ( x  e.  A  /\  y  e.  A
)  /\  ( F `  x )  =  ( F `  y ) ) )  ->  x  =  y )
63623anassrs 1232 . . . . 5  |-  ( ( ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  /\  ( x  e.  A  /\  y  e.  A ) )  /\  ( F `  x )  =  ( F `  y ) )  ->  x  =  y )
6463ex 436 . . . 4  |-  ( ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  /\  (
x  e.  A  /\  y  e.  A )
)  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
6564ralrimivva 2809 . . 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 6159 . . 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 670 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  ( `' F " {  .0.  } )  =  { N } )  ->  F : A -1-1-> B )
6842, 67impbida 843 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737    C_ wss 3404   {csn 3968   `'ccnv 4833   "cima 4837    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   ` cfv 5582  (class class class)co 6290   Basecbs 15121   0gc0g 15338   Grpcgrp 16669   -gcsg 16671    GrpHom cghm 16880   Ringcrg 17780   RingHom crh 17940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-grp 16673  df-minusg 16674  df-sbg 16675  df-ghm 16881  df-mgp 17724  df-ur 17736  df-ring 17782  df-rnghom 17943
This theorem is referenced by:  zrhf1ker  28779
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