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Theorem mulgrhm2 18711
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 18689 . . . . . . . . . 10  |-  ZZ  =  ( Base ` ring )
2 eqid 2454 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
31, 2rhmf 17570 . . . . . . . . 9  |-  ( f  e.  (ring RingHom  R )  ->  f : ZZ --> ( Base `  R
) )
43adantl 464 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f : ZZ --> ( Base `  R ) )
54feqmptd 5901 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  =  ( n  e.  ZZ  |->  ( f `
 n ) ) )
6 rhmghm 17569 . . . . . . . . . . 11  |-  ( f  e.  (ring RingHom  R )  ->  f  e.  (ring  GrpHom  R ) )
76ad2antlr 724 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  f  e.  (ring  GrpHom  R ) )
8 simpr 459 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
9 1zzd 10891 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  1  e.  ZZ )
10 eqid 2454 . . . . . . . . . . 11  |-  (.g ` ring )  =  (.g ` ring )
11 mulgghm2.m . . . . . . . . . . 11  |-  .x.  =  (.g
`  R )
121, 10, 11ghmmulg 16478 . . . . . . . . . 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 1226 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  ( n (.g ` ring ) 1 ) )  =  ( n  .x.  ( f `  1
) ) )
14 ax-1cn 9539 . . . . . . . . . . . . 13  |-  1  e.  CC
15 cnfldmulg 18645 . . . . . . . . . . . . 13  |-  ( ( n  e.  ZZ  /\  1  e.  CC )  ->  ( n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
1614, 15mpan2 669 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
17 1z 10890 . . . . . . . . . . . . 13  |-  1  e.  ZZ
1816adantr 463 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
19 zringmulg 18691 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` ring ) 1 )  =  ( n  x.  1 ) )
2018, 19eqtr4d 2498 . . . . . . . . . . . . 13  |-  ( ( n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n (.g ` ring ) 1 ) )
2117, 20mpan2 669 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n (.g ` ring ) 1 ) )
22 zcn 10865 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  CC )
2322mulid1d 9602 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n  x.  1 )  =  n )
2416, 21, 233eqtr3d 2503 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
n (.g ` ring ) 1 )  =  n )
2524adantl 464 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n (.g ` ring ) 1 )  =  n )
2625fveq2d 5852 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  ( n (.g ` ring ) 1 ) )  =  ( f `  n ) )
27 zring1 18694 . . . . . . . . . . . 12  |-  1  =  ( 1r ` ring )
28 mulgrhm.1 . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  R )
2927, 28rhm1 17574 . . . . . . . . . . 11  |-  ( f  e.  (ring RingHom  R )  ->  (
f `  1 )  =  .1.  )
3029ad2antlr 724 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f ` 
1 )  =  .1.  )
3130oveq2d 6286 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n  .x.  ( f `  1
) )  =  ( n  .x.  .1.  )
)
3213, 26, 313eqtr3d 2503 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  n )  =  ( n  .x.  .1.  )
)
3332mpteq2dva 4525 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
( n  e.  ZZ  |->  ( f `  n
) )  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
345, 33eqtrd 2495 . . . . . 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 2513 . . . . 5  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  =  F )
37 elsn 4030 . . . . 5  |-  ( f  e.  { F }  <->  f  =  F )
3836, 37sylibr 212 . . . 4  |-  ( ( R  e.  Ring  /\  f  e.  (ring RingHom  R ) )  -> 
f  e.  { F } )
3938ex 432 . . 3  |-  ( R  e.  Ring  ->  ( f  e.  (ring RingHom  R )  ->  f  e.  { F } ) )
4039ssrdv 3495 . 2  |-  ( R  e.  Ring  ->  (ring RingHom  R )  C_  { F } )
4111, 35, 28mulgrhm 18710 . . 3  |-  ( R  e.  Ring  ->  F  e.  (ring RingHom  R ) )
4241snssd 4161 . 2  |-  ( R  e.  Ring  ->  { F }  C_  (ring RingHom  R ) )
4340, 42eqssd 3506 1  |-  ( R  e.  Ring  ->  (ring RingHom  R )  =  { F } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {csn 4016    |-> cmpt 4497   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   1c1 9482    x. cmul 9486   ZZcz 10860   Basecbs 14716  .gcmg 16255    GrpHom cghm 16463   1rcur 17348   Ringcrg 17393   RingHom crh 17556  ℂfldccnfld 18615  ℤringzring 18683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-seq 12090  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-grp 16256  df-minusg 16257  df-mulg 16259  df-subg 16397  df-ghm 16464  df-cmn 16999  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-rnghom 17559  df-subrg 17622  df-cnfld 18616  df-zring 18684
This theorem is referenced by:  zrhval2  18721  zrhrhmb  18723  irinitoringc  33131
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