MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgrhm2 Structured version   Unicode version

Theorem mulgrhm2 18038
Description: The powers of the element  1 give the unique ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
Hypotheses
Ref Expression
mulgghm2.m  |-  .x.  =  (.g
`  R )
mulgghm2.f  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
mulgrhm.1  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mulgrhm2  |-  ( R  e.  Ring  ->  (ring RingHom  R )  =  { F } )
Distinct variable groups:    R, n    .x. ,
n    .1. , n
Allowed substitution hint:    F( n)

Proof of Theorem mulgrhm2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 zringbas 18000 . . . . . . . . . 10  |-  ZZ  =  ( Base ` ring )
2 eqid 2451 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
31, 2rhmf 16924 . . . . . . . . 9  |-  ( f  e.  (ring RingHom  R )  ->  f : ZZ --> ( Base `  R
) )
43adantl 466 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f : ZZ --> ( Base `  R ) )
54feqmptd 5845 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  =  ( n  e.  ZZ  |->  ( f `
 n ) ) )
6 rhmghm 16923 . . . . . . . . . . 11  |-  ( f  e.  (ring RingHom  R )  ->  f  e.  (ring  GrpHom  R ) )
76ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  f  e.  (ring  GrpHom  R ) )
8 simpr 461 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
9 1zzd 10780 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  1  e.  ZZ )
10 eqid 2451 . . . . . . . . . . 11  |-  (.g ` ring )  =  (.g ` ring )
11 mulgghm2.m . . . . . . . . . . 11  |-  .x.  =  (.g
`  R )
121, 10, 11ghmmulg 15863 . . . . . . . . . 10  |-  ( ( f  e.  (ring  GrpHom  R )  /\  n  e.  ZZ  /\  1  e.  ZZ )  ->  ( f `  ( n (.g ` ring ) 1 ) )  =  ( n  .x.  ( f `  1
) ) )
137, 8, 9, 12syl3anc 1219 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  ( n (.g ` ring ) 1 ) )  =  ( n  .x.  ( f `  1
) ) )
14 ax-1cn 9443 . . . . . . . . . . . . 13  |-  1  e.  CC
15 cnfldmulg 17959 . . . . . . . . . . . . 13  |-  ( ( n  e.  ZZ  /\  1  e.  CC )  ->  ( n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
1614, 15mpan2 671 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
17 1z 10779 . . . . . . . . . . . . 13  |-  1  e.  ZZ
1816adantr 465 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
19 zringmulg 18002 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` ring ) 1 )  =  ( n  x.  1 ) )
2018, 19eqtr4d 2495 . . . . . . . . . . . . 13  |-  ( ( n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n (.g ` ring ) 1 ) )
2117, 20mpan2 671 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n (.g ` ring ) 1 ) )
22 zcn 10754 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  CC )
2322mulid1d 9506 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n  x.  1 )  =  n )
2416, 21, 233eqtr3d 2500 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
n (.g ` ring ) 1 )  =  n )
2524adantl 466 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n (.g ` ring ) 1 )  =  n )
2625fveq2d 5795 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  ( n (.g ` ring ) 1 ) )  =  ( f `  n ) )
27 zring1 18005 . . . . . . . . . . . 12  |-  1  =  ( 1r ` ring )
28 mulgrhm.1 . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  R )
2927, 28rhm1 16928 . . . . . . . . . . 11  |-  ( f  e.  (ring RingHom  R )  ->  (
f `  1 )  =  .1.  )
3029ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f ` 
1 )  =  .1.  )
3130oveq2d 6208 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n  .x.  ( f `  1
) )  =  ( n  .x.  .1.  )
)
3213, 26, 313eqtr3d 2500 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  n )  =  ( n  .x.  .1.  )
)
3332mpteq2dva 4478 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
( n  e.  ZZ  |->  ( f `  n
) )  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
345, 33eqtrd 2492 . . . . . 6  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  =  ( n  e.  ZZ  |->  ( n 
.x.  .1.  ) )
)
35 mulgghm2.f . . . . . 6  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
3634, 35syl6eqr 2510 . . . . 5  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  =  F )
37 elsn 3991 . . . . 5  |-  ( f  e.  { F }  <->  f  =  F )
3836, 37sylibr 212 . . . 4  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  e.  { F } )
3938ex 434 . . 3  |-  ( R  e.  Ring  ->  ( f  e.  (ring RingHom  R )  ->  f  e.  { F } ) )
4039ssrdv 3462 . 2  |-  ( R  e.  Ring  ->  (ring RingHom  R )  C_  { F } )
4111, 35, 28mulgrhm 18037 . . 3  |-  ( R  e.  Ring  ->  F  e.  (ring RingHom  R ) )
4241snssd 4118 . 2  |-  ( R  e.  Ring  ->  { F }  C_  (ring RingHom  R ) )
4340, 42eqssd 3473 1  |-  ( R  e.  Ring  ->  (ring RingHom  R )  =  { F } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3977    |-> cmpt 4450   -->wf 5514   ` cfv 5518  (class class class)co 6192   CCcc 9383   1c1 9386    x. cmul 9390   ZZcz 10749   Basecbs 14278  .gcmg 15518    GrpHom cghm 15848   1rcur 16710   Ringcrg 16753   RingHom crh 16912  ℂfldccnfld 17929  ℤringzring 17994
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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-addf 9464  ax-mulf 9465
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 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-fz 11541  df-seq 11910  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-starv 14357  df-tset 14361  df-ple 14362  df-ds 14364  df-unif 14365  df-0g 14484  df-mnd 15519  df-mhm 15568  df-grp 15649  df-minusg 15650  df-mulg 15652  df-subg 15782  df-ghm 15849  df-cmn 16385  df-mgp 16699  df-ur 16711  df-rng 16755  df-cring 16756  df-rnghom 16914  df-subrg 16971  df-cnfld 17930  df-zring 17995
This theorem is referenced by:  zrhval2  18051  zrhrhmb  18053
  Copyright terms: Public domain W3C validator