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Theorem kerunit 26425
Description: If a unit element lies in the kernel of a ring homomorphism, then  0  = 
1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.)
Hypotheses
Ref Expression
kerunit.1  |-  U  =  (Unit `  R )
kerunit.2  |-  .0.  =  ( 0g `  S )
kerunit.3  |-  .1.  =  ( 1r `  S )
Assertion
Ref Expression
kerunit  |-  ( ( F  e.  ( R RingHom  S )  /\  ( U  i^i  ( `' F " {  .0.  } ) )  =/=  (/) )  ->  .1.  =  .0.  )

Proof of Theorem kerunit
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3637 . . . . . . . 8  |-  ( x  e.  ( U  i^i  ( `' F " {  .0.  } ) )  <->  ( x  e.  U  /\  x  e.  ( `' F " {  .0.  } ) ) )
21biimpi 194 . . . . . . 7  |-  ( x  e.  ( U  i^i  ( `' F " {  .0.  } ) )  ->  (
x  e.  U  /\  x  e.  ( `' F " {  .0.  }
) ) )
32adantl 466 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( x  e.  U  /\  x  e.  ( `' F " {  .0.  } ) ) )
43simpld 459 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  x  e.  U )
5 rhmrcl1 16915 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
6 kerunit.1 . . . . . . . 8  |-  U  =  (Unit `  R )
7 eqid 2451 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
8 eqid 2451 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2451 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
106, 7, 8, 9unitlinv 16875 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) x )  =  ( 1r `  R
) )
1110fveq2d 5793 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( F `  ( (
( invr `  R ) `  x ) ( .r
`  R ) x ) )  =  ( F `  ( 1r
`  R ) ) )
125, 11sylan 471 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  U )  ->  ( F `  ( (
( invr `  R ) `  x ) ( .r
`  R ) x ) )  =  ( F `  ( 1r
`  R ) ) )
134, 12syldan 470 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( F `  (
( ( invr `  R
) `  x )
( .r `  R
) x ) )  =  ( F `  ( 1r `  R ) ) )
14 simpl 457 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  F  e.  ( R RingHom  S ) )
155adantr 465 . . . . . . 7  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  R  e.  Ring )
16 eqid 2451 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
176, 7, 16rnginvcl 16874 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( invr `  R ) `  x )  e.  (
Base `  R )
)
1815, 4, 17syl2anc 661 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( ( invr `  R
) `  x )  e.  ( Base `  R
) )
1916, 6unitcl 16857 . . . . . . 7  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
204, 19syl 16 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  x  e.  ( Base `  R ) )
21 eqid 2451 . . . . . . 7  |-  ( .r
`  S )  =  ( .r `  S
)
2216, 8, 21rhmmul 16923 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( invr `  R ) `  x )  e.  (
Base `  R )  /\  x  e.  ( Base `  R ) )  ->  ( F `  ( ( ( invr `  R ) `  x
) ( .r `  R ) x ) )  =  ( ( F `  ( (
invr `  R ) `  x ) ) ( .r `  S ) ( F `  x
) ) )
2314, 18, 20, 22syl3anc 1219 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( F `  (
( ( invr `  R
) `  x )
( .r `  R
) x ) )  =  ( ( F `
 ( ( invr `  R ) `  x
) ) ( .r
`  S ) ( F `  x ) ) )
243simprd 463 . . . . . . . 8  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  x  e.  ( `' F " {  .0.  }
) )
25 eqid 2451 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
2616, 25rhmf 16922 . . . . . . . . . 10  |-  ( F  e.  ( R RingHom  S
)  ->  F :
( Base `  R ) --> ( Base `  S )
)
27 ffn 5657 . . . . . . . . . 10  |-  ( F : ( Base `  R
) --> ( Base `  S
)  ->  F  Fn  ( Base `  R )
)
28 elpreima 5922 . . . . . . . . . 10  |-  ( F  Fn  ( Base `  R
)  ->  ( x  e.  ( `' F " {  .0.  } )  <->  ( x  e.  ( Base `  R
)  /\  ( F `  x )  e.  {  .0.  } ) ) )
2926, 27, 283syl 20 . . . . . . . . 9  |-  ( F  e.  ( R RingHom  S
)  ->  ( x  e.  ( `' F " {  .0.  } )  <->  ( x  e.  ( Base `  R
)  /\  ( F `  x )  e.  {  .0.  } ) ) )
3029simplbda 624 . . . . . . . 8  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( `' F " {  .0.  } ) )  ->  ( F `  x )  e.  {  .0.  } )
3124, 30syldan 470 . . . . . . 7  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( F `  x
)  e.  {  .0.  } )
32 fvex 5799 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
3332elsnc 3999 . . . . . . 7  |-  ( ( F `  x )  e.  {  .0.  }  <->  ( F `  x )  =  .0.  )
3431, 33sylib 196 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( F `  x
)  =  .0.  )
3534oveq2d 6206 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( ( F `  ( ( invr `  R
) `  x )
) ( .r `  S ) ( F `
 x ) )  =  ( ( F `
 ( ( invr `  R ) `  x
) ) ( .r
`  S )  .0.  ) )
36 rhmrcl2 16916 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
3736adantr 465 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  S  e.  Ring )
3826adantr 465 . . . . . . 7  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  F : ( Base `  R
) --> ( Base `  S
) )
3938, 18ffvelrnd 5943 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( F `  (
( invr `  R ) `  x ) )  e.  ( Base `  S
) )
40 kerunit.2 . . . . . . 7  |-  .0.  =  ( 0g `  S )
4125, 21, 40rngrz 16788 . . . . . 6  |-  ( ( S  e.  Ring  /\  ( F `  ( ( invr `  R ) `  x ) )  e.  ( Base `  S
) )  ->  (
( F `  (
( invr `  R ) `  x ) ) ( .r `  S )  .0.  )  =  .0.  )
4237, 39, 41syl2anc 661 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( ( F `  ( ( invr `  R
) `  x )
) ( .r `  S )  .0.  )  =  .0.  )
4323, 35, 423eqtrd 2496 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( F `  (
( ( invr `  R
) `  x )
( .r `  R
) x ) )  =  .0.  )
44 kerunit.3 . . . . . 6  |-  .1.  =  ( 1r `  S )
459, 44rhm1 16926 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 1r `  R
) )  =  .1.  )
4645adantr 465 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( F `  ( 1r `  R ) )  =  .1.  )
4713, 43, 463eqtr3rd 2501 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  .1.  =  .0.  )
4847reximdva0 3746 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  ( U  i^i  ( `' F " {  .0.  } ) )  =/=  (/) )  ->  E. x  e.  ( U  i^i  ( `' F " {  .0.  } ) )  .1.  =  .0.  )
49 id 22 . . 3  |-  (  .1.  =  .0.  ->  .1.  =  .0.  )
5049rexlimivw 2933 . 2  |-  ( E. x  e.  ( U  i^i  ( `' F " {  .0.  } ) )  .1.  =  .0. 
->  .1.  =  .0.  )
5148, 50syl 16 1  |-  ( ( F  e.  ( R RingHom  S )  /\  ( U  i^i  ( `' F " {  .0.  } ) )  =/=  (/) )  ->  .1.  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796    i^i cin 3425   (/)c0 3735   {csn 3975   `'ccnv 4937   "cima 4941    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   Basecbs 14276   .rcmulr 14341   0gc0g 14480   1rcur 16708   Ringcrg 16751  Unitcui 16837   invrcinvr 16869   RingHom crh 16910
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-tpos 6845  df-recs 6932  df-rdg 6966  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-0g 14482  df-mnd 15517  df-mhm 15566  df-grp 15647  df-minusg 15648  df-ghm 15847  df-mgp 16697  df-ur 16709  df-rng 16753  df-oppr 16821  df-dvdsr 16839  df-unit 16840  df-invr 16870  df-rnghom 16912
This theorem is referenced by: (None)
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