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Theorem expghmOLD 17924
Description: Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) Obsolete version of expghm 17923 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 11889 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  ( CC  \  {
0 } ) )
213expa 1187 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  ( CC  \  { 0 } ) )
3 eqid 2443 . . 3  |-  ( x  e.  ZZ  |->  ( A ^ x ) )  =  ( x  e.  ZZ  |->  ( A ^
x ) )
42, 3fmptd 5867 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( x  e.  ZZ  |->  ( A ^ x ) ) : ZZ --> ( CC 
\  { 0 } ) )
5 expaddz 11908 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( A ^ (
y  +  z ) )  =  ( ( A ^ y )  x.  ( A ^
z ) ) )
6 zaddcl 10685 . . . . . 6  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  +  z )  e.  ZZ )
76adantl 466 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( y  +  z )  e.  ZZ )
8 oveq2 6099 . . . . . 6  |-  ( x  =  ( y  +  z )  ->  ( A ^ x )  =  ( A ^ (
y  +  z ) ) )
9 ovex 6116 . . . . . 6  |-  ( A ^ ( y  +  z ) )  e. 
_V
108, 3, 9fvmpt 5774 . . . . 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 6099 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
13 ovex 6116 . . . . . . 7  |-  ( A ^ y )  e. 
_V
1412, 3, 13fvmpt 5774 . . . . . 6  |-  ( y  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  y )  =  ( A ^
y ) )
15 oveq2 6099 . . . . . . 7  |-  ( x  =  z  ->  ( A ^ x )  =  ( A ^ z
) )
16 ovex 6116 . . . . . . 7  |-  ( A ^ z )  e. 
_V
1715, 3, 16fvmpt 5774 . . . . . 6  |-  ( z  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  z )  =  ( A ^
z ) )
1814, 17oveqan12d 6110 . . . . 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 466 . . . 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 2485 . . 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 2808 . 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 17866 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
23 subrgsubg 16871 . . . . . 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 15684 . . . . 5  |-  ( ZZ  e.  (SubGrp ` fld )  ->  Z  e. 
Grp )
2724, 26ax-mp 5 . . . 4  |-  Z  e. 
Grp
28 cnrng 17838 . . . . 5  |-fld  e.  Ring
29 cnfldbas 17822 . . . . . . 7  |-  CC  =  ( Base ` fld )
30 cnfld0 17840 . . . . . . 7  |-  0  =  ( 0g ` fld )
31 cndrng 17845 . . . . . . 7  |-fld  e.  DivRing
3229, 30, 31drngui 16838 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
33 expghmOLD.3 . . . . . . 7  |-  U  =  ( Ms  ( CC  \  { 0 } ) )
34 expghmOLD.2 . . . . . . . 8  |-  M  =  (mulGrp ` fld )
3534oveq1i 6101 . . . . . . 7  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3633, 35eqtri 2463 . . . . . 6  |-  U  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3732, 36unitgrp 16759 . . . . 5  |-  (fld  e.  Ring  ->  U  e.  Grp )
3828, 37ax-mp 5 . . . 4  |-  U  e. 
Grp
3927, 38pm3.2i 455 . . 3  |-  ( Z  e.  Grp  /\  U  e.  Grp )
4025subrgbas 16874 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
4122, 40ax-mp 5 . . . 4  |-  ZZ  =  ( Base `  Z )
42 difss 3483 . . . . 5  |-  ( CC 
\  { 0 } )  C_  CC
4334, 29mgpbas 16597 . . . . . 6  |-  CC  =  ( Base `  M )
4433, 43ressbas2 14229 . . . . 5  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( CC  \  {
0 } )  =  ( Base `  U
) )
4542, 44ax-mp 5 . . . 4  |-  ( CC 
\  { 0 } )  =  ( Base `  U )
46 cnfldadd 17823 . . . . . 6  |-  +  =  ( +g  ` fld )
4725, 46ressplusg 14280 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
4822, 47ax-mp 5 . . . 4  |-  +  =  ( +g  `  Z )
49 fvex 5701 . . . . . 6  |-  (Unit ` fld )  e.  _V
5032, 49eqeltri 2513 . . . . 5  |-  ( CC 
\  { 0 } )  e.  _V
51 cnfldmul 17824 . . . . . . 7  |-  x.  =  ( .r ` fld )
5234, 51mgpplusg 16595 . . . . . 6  |-  x.  =  ( +g  `  M )
5333, 52ressplusg 14280 . . . . 5  |-  ( ( CC  \  { 0 } )  e.  _V  ->  x.  =  ( +g  `  U ) )
5450, 53ax-mp 5 . . . 4  |-  x.  =  ( +g  `  U )
5541, 45, 48, 54isghm 15747 . . 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 909 . 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 664 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   _Vcvv 2972    \ cdif 3325    C_ wss 3328   {csn 3877    e. cmpt 4350   -->wf 5414   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282    + caddc 9285    x. cmul 9287   ZZcz 10646   ^cexp 11865   Basecbs 14174   ↾s cress 14175   +g cplusg 14238   Grpcgrp 15410  SubGrpcsubg 15675    GrpHom cghm 15744  mulGrpcmgp 16591   Ringcrg 16645  Unitcui 16731  SubRingcsubrg 16861  ℂfldccnfld 17818
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-addf 9361  ax-mulf 9362
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-seq 11807  df-exp 11866  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-subg 15678  df-ghm 15745  df-cmn 16279  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-subrg 16863  df-cnfld 17819
This theorem is referenced by: (None)
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