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Theorem mulgrhm2OLD 17935
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 17932 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 17871 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
2 mulgghm2OLD.1 . . . . . . . . . . . 12  |-  Z  =  (flds  ZZ )
32subrgbas 16879 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
41, 3ax-mp 5 . . . . . . . . . 10  |-  ZZ  =  ( Base `  Z )
5 eqid 2443 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
64, 5rhmf 16821 . . . . . . . . 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 5749 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  ( n  e.  ZZ  |->  ( f `  n
) ) )
9 rhmghm 16820 . . . . . . . . . . 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 10682 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  1  e.  ZZ )
13 eqid 2443 . . . . . . . . . . 11  |-  (.g `  Z
)  =  (.g `  Z
)
14 mulgghm2OLD.2 . . . . . . . . . . 11  |-  .x.  =  (.g
`  R )
154, 13, 14ghmmulg 15764 . . . . . . . . . 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 1218 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  (
n (.g `  Z ) 1 ) )  =  ( n  .x.  ( f `
 1 ) ) )
17 ax-1cn 9345 . . . . . . . . . . . . 13  |-  1  e.  CC
18 cnfldmulg 17853 . . . . . . . . . . . . 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 16876 . . . . . . . . . . . . . 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 10682 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  1  e.  ZZ )
24 eqid 2443 . . . . . . . . . . . . . 14  |-  (.g ` fld )  =  (.g ` fld )
2524, 2, 13subgmulg 15700 . . . . . . . . . . . . 13  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  n  e.  ZZ  /\  1  e.  ZZ )  ->  ( n (.g ` fld ) 1 )  =  ( n (.g `  Z ) 1 ) )
2621, 22, 23, 25syl3anc 1218 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n (.g ` fld ) 1 )  =  ( n (.g `  Z
) 1 ) )
27 zcn 10656 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  CC )
2827mulid1d 9408 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  (
n  x.  1 )  =  n )
2919, 26, 283eqtr3d 2483 . . . . . . . . . . 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 5700 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  (
n (.g `  Z ) 1 ) )  =  ( f `  n ) )
32 cnfld1 17846 . . . . . . . . . . . . . 14  |-  1  =  ( 1r ` fld )
332, 32subrg1 16880 . . . . . . . . . . . . 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 16825 . . . . . . . . . . 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 6112 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( n  .x.  (
f `  1 )
)  =  ( n 
.x.  .1.  ) )
3916, 31, 383eqtr3d 2483 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R ) )  /\  n  e.  ZZ )  ->  ( f `  n
)  =  ( n 
.x.  .1.  ) )
4039mpteq2dva 4383 . . . . . . 7  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  ( n  e.  ZZ  |->  ( f `  n ) )  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
) )
418, 40eqtrd 2475 . . . . . 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 2493 . . . . 5  |-  ( ( R  e.  Ring  /\  f  e.  ( Z RingHom  R )
)  ->  f  =  F )
44 elsn 3896 . . . . 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 3367 . 2  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  C_  { F } )
482, 14, 42, 35mulgrhmOLD 17934 . . 3  |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R )
)
4948snssd 4023 . 2  |-  ( R  e.  Ring  ->  { F }  C_  ( Z RingHom  R
) )
5047, 49eqssd 3378 1  |-  ( R  e.  Ring  ->  ( Z RingHom  R )  =  { F } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3882    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   1c1 9288    x. cmul 9292   ZZcz 10651   Basecbs 14179   ↾s cress 14180  .gcmg 15419  SubGrpcsubg 15680    GrpHom cghm 15749   1rcur 16608   Ringcrg 16650   RingHom crh 16809  SubRingcsubrg 16866  ℂfldccnfld 17823
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-addf 9366  ax-mulf 9367
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-seq 11812  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-0g 14385  df-mnd 15420  df-mhm 15469  df-grp 15550  df-minusg 15551  df-mulg 15553  df-subg 15683  df-ghm 15750  df-cmn 16284  df-mgp 16597  df-ur 16609  df-rng 16652  df-cring 16653  df-rnghom 16811  df-subrg 16868  df-cnfld 17824
This theorem is referenced by: (None)
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