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Theorem expghmOLD 17883
Description: Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) Obsolete version of expghm 17882 as of 10-Jun-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
expghmOLD.1  |-  Z  =  (flds  ZZ )
expghmOLD.2  |-  M  =  (mulGrp ` fld )
expghmOLD.3  |-  U  =  ( Ms  ( CC  \  { 0 } ) )
Assertion
Ref Expression
expghmOLD  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( x  e.  ZZ  |->  ( A ^ x ) )  e.  ( Z 
GrpHom  U ) )
Distinct variable group:    x, A
Allowed substitution hints:    U( x)    M( x)    Z( x)

Proof of Theorem expghmOLD
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 expclzlem 11885 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  ( CC  \  {
0 } ) )
213expa 1182 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  ( CC  \  { 0 } ) )
3 eqid 2441 . . 3  |-  ( x  e.  ZZ  |->  ( A ^ x ) )  =  ( x  e.  ZZ  |->  ( A ^
x ) )
42, 3fmptd 5864 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( x  e.  ZZ  |->  ( A ^ x ) ) : ZZ --> ( CC 
\  { 0 } ) )
5 expaddz 11904 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( A ^ (
y  +  z ) )  =  ( ( A ^ y )  x.  ( A ^
z ) ) )
6 zaddcl 10681 . . . . . 6  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  +  z )  e.  ZZ )
76adantl 463 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( y  +  z )  e.  ZZ )
8 oveq2 6098 . . . . . 6  |-  ( x  =  ( y  +  z )  ->  ( A ^ x )  =  ( A ^ (
y  +  z ) ) )
9 ovex 6115 . . . . . 6  |-  ( A ^ ( y  +  z ) )  e. 
_V
108, 3, 9fvmpt 5771 . . . . 5  |-  ( ( y  +  z )  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  ( y  +  z ) )  =  ( A ^
( y  +  z ) ) )
117, 10syl 16 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( ( x  e.  ZZ  |->  ( A ^
x ) ) `  ( y  +  z ) )  =  ( A ^ ( y  +  z ) ) )
12 oveq2 6098 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
13 ovex 6115 . . . . . . 7  |-  ( A ^ y )  e. 
_V
1412, 3, 13fvmpt 5771 . . . . . 6  |-  ( y  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  y )  =  ( A ^
y ) )
15 oveq2 6098 . . . . . . 7  |-  ( x  =  z  ->  ( A ^ x )  =  ( A ^ z
) )
16 ovex 6115 . . . . . . 7  |-  ( A ^ z )  e. 
_V
1715, 3, 16fvmpt 5771 . . . . . 6  |-  ( z  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  z )  =  ( A ^
z ) )
1814, 17oveqan12d 6109 . . . . 5  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( ( x  e.  ZZ  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e.  ZZ  |->  ( A ^
x ) ) `  z ) )  =  ( ( A ^
y )  x.  ( A ^ z ) ) )
1918adantl 463 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( ( ( x  e.  ZZ  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e.  ZZ  |->  ( A ^
x ) ) `  z ) )  =  ( ( A ^
y )  x.  ( A ^ z ) ) )
205, 11, 193eqtr4d 2483 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( ( x  e.  ZZ  |->  ( A ^
x ) ) `  ( y  +  z ) )  =  ( ( ( x  e.  ZZ  |->  ( A ^
x ) ) `  y )  x.  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  z ) ) )
2120ralrimivva 2806 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A. y  e.  ZZ  A. z  e.  ZZ  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  ( y  +  z ) )  =  ( ( ( x  e.  ZZ  |->  ( A ^ x ) ) `  y )  x.  ( ( x  e.  ZZ  |->  ( A ^ x ) ) `
 z ) ) )
22 zsubrg 17825 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
23 subrgsubg 16851 . . . . . 6  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
2422, 23ax-mp 5 . . . . 5  |-  ZZ  e.  (SubGrp ` fld )
25 expghmOLD.1 . . . . . 6  |-  Z  =  (flds  ZZ )
2625subggrp 15677 . . . . 5  |-  ( ZZ  e.  (SubGrp ` fld )  ->  Z  e. 
Grp )
2724, 26ax-mp 5 . . . 4  |-  Z  e. 
Grp
28 cnrng 17797 . . . . 5  |-fld  e.  Ring
29 cnfldbas 17781 . . . . . . 7  |-  CC  =  ( Base ` fld )
30 cnfld0 17799 . . . . . . 7  |-  0  =  ( 0g ` fld )
31 cndrng 17804 . . . . . . 7  |-fld  e.  DivRing
3229, 30, 31drngui 16818 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
33 expghmOLD.3 . . . . . . 7  |-  U  =  ( Ms  ( CC  \  { 0 } ) )
34 expghmOLD.2 . . . . . . . 8  |-  M  =  (mulGrp ` fld )
3534oveq1i 6100 . . . . . . 7  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3633, 35eqtri 2461 . . . . . 6  |-  U  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3732, 36unitgrp 16749 . . . . 5  |-  (fld  e.  Ring  ->  U  e.  Grp )
3828, 37ax-mp 5 . . . 4  |-  U  e. 
Grp
3927, 38pm3.2i 452 . . 3  |-  ( Z  e.  Grp  /\  U  e.  Grp )
4025subrgbas 16854 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
4122, 40ax-mp 5 . . . 4  |-  ZZ  =  ( Base `  Z )
42 difss 3480 . . . . 5  |-  ( CC 
\  { 0 } )  C_  CC
4334, 29mgpbas 16587 . . . . . 6  |-  CC  =  ( Base `  M )
4433, 43ressbas2 14225 . . . . 5  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( CC  \  {
0 } )  =  ( Base `  U
) )
4542, 44ax-mp 5 . . . 4  |-  ( CC 
\  { 0 } )  =  ( Base `  U )
46 cnfldadd 17782 . . . . . 6  |-  +  =  ( +g  ` fld )
4725, 46ressplusg 14276 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
4822, 47ax-mp 5 . . . 4  |-  +  =  ( +g  `  Z )
49 fvex 5698 . . . . . 6  |-  (Unit ` fld )  e.  _V
5032, 49eqeltri 2511 . . . . 5  |-  ( CC 
\  { 0 } )  e.  _V
51 cnfldmul 17783 . . . . . . 7  |-  x.  =  ( .r ` fld )
5234, 51mgpplusg 16585 . . . . . 6  |-  x.  =  ( +g  `  M )
5333, 52ressplusg 14276 . . . . 5  |-  ( ( CC  \  { 0 } )  e.  _V  ->  x.  =  ( +g  `  U ) )
5450, 53ax-mp 5 . . . 4  |-  x.  =  ( +g  `  U )
5541, 45, 48, 54isghm 15740 . . 3  |-  ( ( x  e.  ZZ  |->  ( A ^ x ) )  e.  ( Z 
GrpHom  U )  <->  ( ( Z  e.  Grp  /\  U  e.  Grp )  /\  (
( x  e.  ZZ  |->  ( A ^ x ) ) : ZZ --> ( CC 
\  { 0 } )  /\  A. y  e.  ZZ  A. z  e.  ZZ  ( ( x  e.  ZZ  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( ( ( x  e.  ZZ  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e.  ZZ  |->  ( A ^
x ) ) `  z ) ) ) ) )
5639, 55mpbiran 904 . 2  |-  ( ( x  e.  ZZ  |->  ( A ^ x ) )  e.  ( Z 
GrpHom  U )  <->  ( (
x  e.  ZZ  |->  ( A ^ x ) ) : ZZ --> ( CC 
\  { 0 } )  /\  A. y  e.  ZZ  A. z  e.  ZZ  ( ( x  e.  ZZ  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( ( ( x  e.  ZZ  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e.  ZZ  |->  ( A ^
x ) ) `  z ) ) ) )
574, 21, 56sylanbrc 659 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( x  e.  ZZ  |->  ( A ^ x ) )  e.  ( Z 
GrpHom  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970    \ cdif 3322    C_ wss 3325   {csn 3874    e. cmpt 4347   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278    + caddc 9281    x. cmul 9283   ZZcz 10642   ^cexp 11861   Basecbs 14170   ↾s cress 14171   +g cplusg 14234   Grpcgrp 15406  SubGrpcsubg 15668    GrpHom cghm 15737  mulGrpcmgp 16581   Ringcrg 16635  Unitcui 16721  SubRingcsubrg 16841  ℂfldccnfld 17777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-seq 11803  df-exp 11862  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-subg 15671  df-ghm 15738  df-cmn 16272  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-subrg 16843  df-cnfld 17778
This theorem is referenced by: (None)
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