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Theorem kerunit 27476
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 3687 . . . . . . . 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 17152 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
6 kerunit.1 . . . . . . . 8  |-  U  =  (Unit `  R )
7 eqid 2467 . . . . . . . 8  |-  ( invr `  R )  =  (
invr `  R )
8 eqid 2467 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2467 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
106, 7, 8, 9unitlinv 17110 . . . . . . 7  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) x )  =  ( 1r `  R
) )
1110fveq2d 5868 . . . . . 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 2467 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
176, 7, 16rnginvcl 17109 . . . . . . 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 17092 . . . . . . 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 2467 . . . . . . 7  |-  ( .r
`  S )  =  ( .r `  S
)
2216, 8, 21rhmmul 17160 . . . . . 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 1228 . . . . 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 2467 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
2616, 25rhmf 17159 . . . . . . . . . 10  |-  ( F  e.  ( R RingHom  S
)  ->  F :
( Base `  R ) --> ( Base `  S )
)
27 ffn 5729 . . . . . . . . . 10  |-  ( F : ( Base `  R
) --> ( Base `  S
)  ->  F  Fn  ( Base `  R )
)
28 elpreima 5999 . . . . . . . . . 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 5874 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
3332elsnc 4051 . . . . . . 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 6298 . . . . 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 17153 . . . . . . 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 6020 . . . . . 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 17023 . . . . . 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 2512 . . . 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 17163 . . . . 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 2517 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( U  i^i  ( `' F " {  .0.  } ) ) )  ->  .1.  =  .0.  )
4847reximdva0 3796 . 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 2952 . 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    i^i cin 3475   (/)c0 3785   {csn 4027   `'ccnv 4998   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   Basecbs 14486   .rcmulr 14552   0gc0g 14691   1rcur 16943   Ringcrg 16986  Unitcui 17072   invrcinvr 17104   RingHom crh 17145
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-0g 14693  df-mnd 15728  df-mhm 15777  df-grp 15858  df-minusg 15859  df-ghm 16060  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-rnghom 17148
This theorem is referenced by: (None)
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