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Theorem kerunit 28279
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 3628 . . . . . . . 8  |-  ( x  e.  ( U  i^i  ( `' F " {  .0.  } ) )  <->  ( x  e.  U  /\  x  e.  ( `' F " {  .0.  } ) ) )
21biimpi 196 . . . . . . 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 17690 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
6 kerunit.1 . . . . . . . 8  |-  U  =  (Unit `  R )
7 eqid 2404 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
8 eqid 2404 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2404 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
106, 7, 8, 9unitlinv 17648 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) x )  =  ( 1r `  R
) )
1110fveq2d 5855 . . . . . 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 2404 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
176, 7, 16ringinvcl 17647 . . . . . . 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 17630 . . . . . . 7  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
204, 19syl 17 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  x  e.  ( Base `  R ) )
21 eqid 2404 . . . . . . 7  |-  ( .r
`  S )  =  ( .r `  S
)
2216, 8, 21rhmmul 17698 . . . . . 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 1232 . . . . 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 2404 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
2616, 25rhmf 17697 . . . . . . . . . 10  |-  ( F  e.  ( R RingHom  S
)  ->  F :
( Base `  R ) --> ( Base `  S )
)
27 ffn 5716 . . . . . . . . . 10  |-  ( F : ( Base `  R
) --> ( Base `  S
)  ->  F  Fn  ( Base `  R )
)
28 elpreima 5987 . . . . . . . . . 10  |-  ( F  Fn  ( Base `  R
)  ->  ( x  e.  ( `' F " {  .0.  } )  <->  ( x  e.  ( Base `  R
)  /\  ( F `  x )  e.  {  .0.  } ) ) )
2926, 27, 283syl 18 . . . . . . . . 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 5861 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
3332elsnc 3998 . . . . . . 7  |-  ( ( F `  x )  e.  {  .0.  }  <->  ( F `  x )  =  .0.  )
3431, 33sylib 198 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  -> 
( F `  x
)  =  .0.  )
3534oveq2d 6296 . . . . 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 17691 . . . . . . 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 6012 . . . . . 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, 40ringrz 17558 . . . . . 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 2449 . . . 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 17701 . . . . 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 2454 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  .1.  =  .0.  )
4847reximdva0 3752 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  ( U  i^i  ( `' F " {  .0.  } ) )  =/=  (/) )  ->  E. x  e.  ( U  i^i  ( `' F " {  .0.  } ) )  .1.  =  .0.  )
49 id 23 . . 3  |-  (  .1.  =  .0.  ->  .1.  =  .0.  )
5049rexlimivw 2895 . 2  |-  ( E. x  e.  ( U  i^i  ( `' F " {  .0.  } ) )  .1.  =  .0. 
->  .1.  =  .0.  )
5148, 50syl 17 1  |-  ( ( F  e.  ( R RingHom  S )  /\  ( U  i^i  ( `' F " {  .0.  } ) )  =/=  (/) )  ->  .1.  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   E.wrex 2757    i^i cin 3415   (/)c0 3740   {csn 3974   `'ccnv 4824   "cima 4828    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280   Basecbs 14843   .rcmulr 14912   0gc0g 15056   1rcur 17475   Ringcrg 17520  Unitcui 17610   invrcinvr 17642   RingHom crh 17683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-tpos 6960  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-0g 15058  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-mhm 16292  df-grp 16383  df-minusg 16384  df-ghm 16591  df-mgp 17464  df-ur 17476  df-ring 17522  df-oppr 17594  df-dvdsr 17612  df-unit 17613  df-invr 17643  df-rnghom 17686
This theorem is referenced by: (None)
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