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Theorem mulgrhm2OLD 18405
Description: The powers of the element  1 give the unique ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) Obsolete version of mulgrhm2 18402 as of 12-Jun-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
mulgghm2OLD.1  |-  Z  =  (flds  ZZ )
mulgghm2OLD.2  |-  .x.  =  (.g
`  R )
mulgghm2OLD.3  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
mulgrhmOLD.4  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mulgrhm2OLD  |-  ( 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 mulgrhm2OLD
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 zsubrg 18341 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
2 mulgghm2OLD.1 . . . . . . . . . . . 12  |-  Z  =  (flds  ZZ )
32subrgbas 17309 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
41, 3ax-mp 5 . . . . . . . . . 10  |-  ZZ  =  ( Base `  Z )
5 eqid 2467 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
64, 5rhmf 17247 . . . . . . . . 9  |-  ( f  e.  ( Z RingHom  R
)  ->  f : ZZ
--> ( Base `  R
) )
76adantl 466 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f : ZZ
--> ( Base `  R
) )
87feqmptd 5927 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  ( n  e.  ZZ  |->  ( f `  n
) ) )
9 rhmghm 17246 . . . . . . . . . . 11  |-  ( f  e.  ( Z RingHom  R
)  ->  f  e.  ( Z  GrpHom  R ) )
109ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  f  e.  ( Z 
GrpHom  R ) )
11 simpr 461 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
12 1zzd 10907 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  1  e.  ZZ )
13 eqid 2467 . . . . . . . . . . 11  |-  (.g `  Z
)  =  (.g `  Z
)
14 mulgghm2OLD.2 . . . . . . . . . . 11  |-  .x.  =  (.g
`  R )
154, 13, 14ghmmulg 16151 . . . . . . . . . 10  |-  ( ( f  e.  ( Z 
GrpHom  R )  /\  n  e.  ZZ  /\  1  e.  ZZ )  ->  (
f `  ( n
(.g `  Z ) 1 ) )  =  ( n  .x.  ( f `
 1 ) ) )
1610, 11, 12, 15syl3anc 1228 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  (
n (.g `  Z ) 1 ) )  =  ( n  .x.  ( f `
 1 ) ) )
17 ax-1cn 9562 . . . . . . . . . . . . 13  |-  1  e.  CC
18 cnfldmulg 18320 . . . . . . . . . . . . 13  |-  ( ( n  e.  ZZ  /\  1  e.  CC )  ->  ( n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
1917, 18mpan2 671 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n  x.  1 ) )
20 subrgsubg 17306 . . . . . . . . . . . . . 14  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
211, 20mp1i 12 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  ZZ  e.  (SubGrp ` fld ) )
22 id 22 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  ZZ )
23 1zzd 10907 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  1  e.  ZZ )
24 eqid 2467 . . . . . . . . . . . . . 14  |-  (.g ` fld )  =  (.g ` fld )
2524, 2, 13subgmulg 16087 . . . . . . . . . . . . 13  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n (.g `  Z ) 1 ) )
2621, 22, 23, 25syl3anc 1228 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n (.g `  Z
) 1 ) )
27 zcn 10881 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  CC )
2827mulid1d 9625 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n  x.  1 )  =  n )
2919, 26, 283eqtr3d 2516 . . . . . . . . . . 11  |-  ( n  e.  ZZ  ->  (
n (.g `  Z ) 1 )  =  n )
3029adantl 466 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n (.g `  Z
) 1 )  =  n )
3130fveq2d 5876 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  (
n (.g `  Z ) 1 ) )  =  ( f `  n ) )
32 cnfld1 18313 . . . . . . . . . . . . . 14  |-  1  =  ( 1r ` fld )
332, 32subrg1 17310 . . . . . . . . . . . . 13  |-  ( ZZ  e.  (SubRing ` fld )  ->  1  =  ( 1r `  Z
) )
341, 33ax-mp 5 . . . . . . . . . . . 12  |-  1  =  ( 1r `  Z )
35 mulgrhmOLD.4 . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  R )
3634, 35rhm1 17251 . . . . . . . . . . 11  |-  ( f  e.  ( Z RingHom  R
)  ->  ( f `  1 )  =  .1.  )
3736ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  1
)  =  .1.  )
3837oveq2d 6311 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n  .x.  (
f `  1 )
)  =  ( n 
.x.  .1.  ) )
3916, 31, 383eqtr3d 2516 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  n
)  =  ( n 
.x.  .1.  ) )
4039mpteq2dva 4539 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  ( n  e.  ZZ  |->  ( f `  n ) )  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
418, 40eqtrd 2508 . . . . . 6  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
42 mulgghm2OLD.3 . . . . . 6  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
4341, 42syl6eqr 2526 . . . . 5  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  F )
44 elsn 4047 . . . . 5  |-  ( f  e.  { F }  <->  f  =  F )
4543, 44sylibr 212 . . . 4  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  e.  { F } )
4645ex 434 . . 3  |-  ( R  e.  Ring  ->  ( f  e.  ( Z RingHom  R
)  ->  f  e.  { F } ) )
4746ssrdv 3515 . 2  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  C_  { F } )
482, 14, 42, 35mulgrhmOLD 18404 . . 3  |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R )
)
4948snssd 4178 . 2  |-  ( R  e.  Ring  ->  { F }  C_  ( Z RingHom  R
) )
5047, 49eqssd 3526 1  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  =  { F } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4033    |-> cmpt 4511   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   1c1 9505    x. cmul 9509   ZZcz 10876   Basecbs 14507   ↾s cress 14508  .gcmg 15928  SubGrpcsubg 16067    GrpHom cghm 16136   1rcur 17025   Ringcrg 17070   RingHom crh 17233  SubRingcsubrg 17296  ℂfldccnfld 18290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-seq 12088  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-grp 15929  df-minusg 15930  df-mulg 15932  df-subg 16070  df-ghm 16137  df-cmn 16673  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-rnghom 17236  df-subrg 17298  df-cnfld 18291
This theorem is referenced by: (None)
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