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Theorem mulgrhm2 17902
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 17864 . . . . . . . . . 10  |-  ZZ  =  ( Base ` ring )
2 eqid 2438 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
31, 2rhmf 16802 . . . . . . . . 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 5739 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  =  ( n  e.  ZZ  |->  ( f `
 n ) ) )
6 rhmghm 16801 . . . . . . . . . . 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 10669 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  1  e.  ZZ )
10 eqid 2438 . . . . . . . . . . 11  |-  (.g ` ring )  =  (.g ` ring )
11 mulgghm2.m . . . . . . . . . . 11  |-  .x.  =  (.g
`  R )
121, 10, 11ghmmulg 15750 . . . . . . . . . 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 1218 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  ( n (.g ` ring ) 1 ) )  =  ( n  .x.  ( f `  1
) ) )
14 ax-1cn 9332 . . . . . . . . . . . . 13  |-  1  e.  CC
15 cnfldmulg 17823 . . . . . . . . . . . . 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 10668 . . . . . . . . . . . . 13  |-  1  e.  ZZ
1816adantr 465 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
19 zringmulg 17866 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` ring ) 1 )  =  ( n  x.  1 ) )
2018, 19eqtr4d 2473 . . . . . . . . . . . . 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 10643 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  CC )
2322mulid1d 9395 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n  x.  1 )  =  n )
2416, 21, 233eqtr3d 2478 . . . . . . . . . . 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 5690 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  ( n (.g ` ring ) 1 ) )  =  ( f `  n ) )
27 zring1 17869 . . . . . . . . . . . 12  |-  1  =  ( 1r ` ring )
28 mulgrhm.1 . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  R )
2927, 28rhm1 16806 . . . . . . . . . . 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 6102 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n  .x.  ( f `  1
) )  =  ( n  .x.  .1.  )
)
3213, 26, 313eqtr3d 2478 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  n )  =  ( n  .x.  .1.  )
)
3332mpteq2dva 4373 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
( n  e.  ZZ  |->  ( f `  n
) )  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
345, 33eqtrd 2470 . . . . . 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 2488 . . . . 5  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  =  F )
37 elsn 3886 . . . . 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 3357 . 2  |-  ( R  e.  Ring  ->  (ring RingHom  R )  C_  { F } )
4111, 35, 28mulgrhm 17901 . . 3  |-  ( R  e.  Ring  ->  F  e.  (ring RingHom  R ) )
4241snssd 4013 . 2  |-  ( R  e.  Ring  ->  { F }  C_  (ring RingHom  R ) )
4340, 42eqssd 3368 1  |-  ( R  e.  Ring  ->  (ring RingHom  R )  =  { F } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3872    e. cmpt 4345   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   1c1 9275    x. cmul 9279   ZZcz 10638   Basecbs 14166  .gcmg 15406    GrpHom cghm 15735   1rcur 16591   Ringcrg 16633   RingHom crh 16792  ℂfldccnfld 17793  ℤringzring 17858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-seq 11799  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-0g 14372  df-mnd 15407  df-mhm 15456  df-grp 15536  df-minusg 15537  df-mulg 15539  df-subg 15669  df-ghm 15736  df-cmn 16270  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-rnghom 16794  df-subrg 16841  df-cnfld 17794  df-zring 17859
This theorem is referenced by:  zrhval2  17915  zrhrhmb  17917
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