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Theorem mulgrhm2 16780
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 16744 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
2 mulgghm2.1 . . . . . . . . . . . 12  |-  Z  =  (flds  ZZ )
32subrgbas 15869 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
41, 3ax-mp 8 . . . . . . . . . 10  |-  ZZ  =  ( Base `  Z )
5 eqid 2435 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
64, 5rhmf 15819 . . . . . . . . 9  |-  ( f  e.  ( Z RingHom  R
)  ->  f : ZZ
--> ( Base `  R
) )
76adantl 453 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f : ZZ
--> ( Base `  R
) )
87feqmptd 5771 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  ( n  e.  ZZ  |->  ( f `  n
) ) )
9 rhmghm 15818 . . . . . . . . . . 11  |-  ( f  e.  ( Z RingHom  R
)  ->  f  e.  ( Z  GrpHom  R ) )
109ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  f  e.  ( Z 
GrpHom  R ) )
11 simpr 448 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
12 1z 10303 . . . . . . . . . . 11  |-  1  e.  ZZ
1312a1i 11 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  1  e.  ZZ )
14 eqid 2435 . . . . . . . . . . 11  |-  (.g `  Z
)  =  (.g `  Z
)
15 mulgghm2.2 . . . . . . . . . . 11  |-  .x.  =  (.g
`  R )
164, 14, 15ghmmulg 15010 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  (
n (.g `  Z ) 1 ) )  =  ( n  .x.  ( f `
 1 ) ) )
18 ax-1cn 9040 . . . . . . . . . . . . 13  |-  1  e.  CC
19 cnfldmulg 16725 . . . . . . . . . . . . 13  |-  ( ( n  e.  ZZ  /\  1  e.  CC )  ->  ( n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
2018, 19mpan2 653 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
21 subrgsubg 15866 . . . . . . . . . . . . . 14  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
221, 21mp1i 12 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  ZZ  e.  (SubGrp ` fld ) )
23 id 20 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  ZZ )
2412a1i 11 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  1  e.  ZZ )
25 eqid 2435 . . . . . . . . . . . . . 14  |-  (.g ` fld )  =  (.g ` fld )
2625, 2, 14subgmulg 14950 . . . . . . . . . . . . 13  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n (.g `  Z ) 1 ) )
2722, 23, 24, 26syl3anc 1184 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n (.g `  Z
) 1 ) )
28 zcn 10279 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  CC )
2928mulid1d 9097 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n  x.  1 )  =  n )
3020, 27, 293eqtr3d 2475 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
n (.g `  Z ) 1 )  =  n )
3130adantl 453 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n (.g `  Z
) 1 )  =  n )
3231fveq2d 5724 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  (
n (.g `  Z ) 1 ) )  =  ( f `  n ) )
33 cnfld1 16718 . . . . . . . . . . . . . 14  |-  1  =  ( 1r ` fld )
342, 33subrg1 15870 . . . . . . . . . . . . 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 15823 . . . . . . . . . . 11  |-  ( f  e.  ( Z RingHom  R
)  ->  ( f `  1 )  =  .1.  )
3837ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  1
)  =  .1.  )
3938oveq2d 6089 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n  .x.  (
f `  1 )
)  =  ( n 
.x.  .1.  ) )
4017, 32, 393eqtr3d 2475 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  n
)  =  ( n 
.x.  .1.  ) )
4140mpteq2dva 4287 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  ( n  e.  ZZ  |->  ( f `  n ) )  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
428, 41eqtrd 2467 . . . . . 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 2485 . . . . 5  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  F )
45 elsn 3821 . . . . 5  |-  ( f  e.  { F }  <->  f  =  F )
4644, 45sylibr 204 . . . 4  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  e.  { F } )
4746ex 424 . . 3  |-  ( R  e.  Ring  ->  ( f  e.  ( Z RingHom  R
)  ->  f  e.  { F } ) )
4847ssrdv 3346 . 2  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  C_  { F } )
492, 15, 43, 36mulgrhm 16779 . . 3  |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R )
)
5049snssd 3935 . 2  |-  ( R  e.  Ring  ->  { F }  C_  ( Z RingHom  R
) )
5148, 50eqssd 3357 1  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  =  { F } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983    x. cmul 8987   ZZcz 10274   Basecbs 13461   ↾s cress 13462  .gcmg 14681  SubGrpcsubg 14930    GrpHom cghm 14995   Ringcrg 15652   1rcur 15654   RingHom crh 15809  SubRingcsubrg 15856  ℂfldccnfld 16695
This theorem is referenced by:  zrhval2  16782  zrhrhmb  16784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-seq 11316  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-mnd 14682  df-mhm 14730  df-grp 14804  df-minusg 14805  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cmn 15406  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-rnghom 15811  df-subrg 15858  df-cnfld 16696
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