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Theorem mulgrhm2 16461
Description: The powers of the element  1 give the unique ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
mulgghm2.1  |-  Z  =  (flds  ZZ )
mulgghm2.2  |-  .x.  =  (.g
`  R )
mulgghm2.3  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
mulgrhm.4  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mulgrhm2  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  =  { F } )
Distinct variable groups:    R, n    .x. ,
n    n, Z    .1. , n
Allowed substitution hint:    F( n)

Proof of Theorem mulgrhm2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 zsubrg 16425 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
2 mulgghm2.1 . . . . . . . . . . . 12  |-  Z  =  (flds  ZZ )
32subrgbas 15554 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
41, 3ax-mp 8 . . . . . . . . . 10  |-  ZZ  =  ( Base `  Z )
5 eqid 2283 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
64, 5rhmf 15504 . . . . . . . . 9  |-  ( f  e.  ( Z RingHom  R
)  ->  f : ZZ
--> ( Base `  R
) )
76adantl 452 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f : ZZ
--> ( Base `  R
) )
87feqmptd 5575 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  ( n  e.  ZZ  |->  ( f `  n
) ) )
9 rhmghm 15503 . . . . . . . . . . 11  |-  ( f  e.  ( Z RingHom  R
)  ->  f  e.  ( Z  GrpHom  R ) )
109ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  f  e.  ( Z 
GrpHom  R ) )
11 simpr 447 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
12 1z 10053 . . . . . . . . . . 11  |-  1  e.  ZZ
1312a1i 10 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  1  e.  ZZ )
14 eqid 2283 . . . . . . . . . . 11  |-  (.g `  Z
)  =  (.g `  Z
)
15 mulgghm2.2 . . . . . . . . . . 11  |-  .x.  =  (.g
`  R )
164, 14, 15ghmmulg 14695 . . . . . . . . . 10  |-  ( ( f  e.  ( Z 
GrpHom  R )  /\  n  e.  ZZ  /\  1  e.  ZZ )  ->  (
f `  ( n
(.g `  Z ) 1 ) )  =  ( n  .x.  ( f `
 1 ) ) )
1710, 11, 13, 16syl3anc 1182 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  (
n (.g `  Z ) 1 ) )  =  ( n  .x.  ( f `
 1 ) ) )
18 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
19 cnfldmulg 16406 . . . . . . . . . . . . 13  |-  ( ( n  e.  ZZ  /\  1  e.  CC )  ->  ( n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
2018, 19mpan2 652 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
21 subrgsubg 15551 . . . . . . . . . . . . . 14  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
221, 21mp1i 11 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  ZZ  e.  (SubGrp ` fld ) )
23 id 19 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  ZZ )
2412a1i 10 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  1  e.  ZZ )
25 eqid 2283 . . . . . . . . . . . . . 14  |-  (.g ` fld )  =  (.g ` fld )
2625, 2, 14subgmulg 14635 . . . . . . . . . . . . 13  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n (.g `  Z ) 1 ) )
2722, 23, 24, 26syl3anc 1182 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n (.g `  Z
) 1 ) )
28 zcn 10029 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  CC )
2928mulid1d 8852 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n  x.  1 )  =  n )
3020, 27, 293eqtr3d 2323 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
n (.g `  Z ) 1 )  =  n )
3130adantl 452 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n (.g `  Z
) 1 )  =  n )
3231fveq2d 5529 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  (
n (.g `  Z ) 1 ) )  =  ( f `  n ) )
33 cnfld1 16399 . . . . . . . . . . . . . 14  |-  1  =  ( 1r ` fld )
342, 33subrg1 15555 . . . . . . . . . . . . 13  |-  ( ZZ  e.  (SubRing ` fld )  ->  1  =  ( 1r `  Z
) )
351, 34ax-mp 8 . . . . . . . . . . . 12  |-  1  =  ( 1r `  Z )
36 mulgrhm.4 . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  R )
3735, 36rhm1 15508 . . . . . . . . . . 11  |-  ( f  e.  ( Z RingHom  R
)  ->  ( f `  1 )  =  .1.  )
3837ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  1
)  =  .1.  )
3938oveq2d 5874 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n  .x.  (
f `  1 )
)  =  ( n 
.x.  .1.  ) )
4017, 32, 393eqtr3d 2323 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  n
)  =  ( n 
.x.  .1.  ) )
4140mpteq2dva 4106 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  ( n  e.  ZZ  |->  ( f `  n ) )  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
428, 41eqtrd 2315 . . . . . 6  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
43 mulgghm2.3 . . . . . 6  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
4442, 43syl6eqr 2333 . . . . 5  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  F )
45 elsn 3655 . . . . 5  |-  ( f  e.  { F }  <->  f  =  F )
4644, 45sylibr 203 . . . 4  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  e.  { F } )
4746ex 423 . . 3  |-  ( R  e.  Ring  ->  ( f  e.  ( Z RingHom  R
)  ->  f  e.  { F } ) )
4847ssrdv 3185 . 2  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  C_  { F } )
492, 15, 43, 36mulgrhm 16460 . . 3  |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R )
)
5049snssd 3760 . 2  |-  ( R  e.  Ring  ->  { F }  C_  ( Z RingHom  R
) )
5148, 50eqssd 3196 1  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  =  { F } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    x. cmul 8742   ZZcz 10024   Basecbs 13148   ↾s cress 13149  .gcmg 14366  SubGrpcsubg 14615    GrpHom cghm 14680   Ringcrg 15337   1rcur 15339   RingHom crh 15494  SubRingcsubrg 15541  ℂfldccnfld 16377
This theorem is referenced by:  zrhval2  16463  zrhrhmb  16465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-seq 11047  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-rnghom 15496  df-subrg 15543  df-cnfld 16378
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