MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  expghm Unicode version

Theorem expghm 16466
Description: Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypotheses
Ref Expression
expghm.1  |-  Z  =  (flds  ZZ )
expghm.2  |-  M  =  (mulGrp ` fld )
expghm.3  |-  U  =  ( Ms  ( CC  \  { 0 } ) )
Assertion
Ref Expression
expghm  |-  ( ( 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 expghm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 expclzlem 11143 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  ( CC  \  {
0 } ) )
213expa 1151 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  ( CC  \  { 0 } ) )
3 eqid 2296 . . 3  |-  ( x  e.  ZZ  |->  ( A ^ x ) )  =  ( x  e.  ZZ  |->  ( A ^
x ) )
42, 3fmptd 5700 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( x  e.  ZZ  |->  ( A ^ x ) ) : ZZ --> ( CC 
\  { 0 } ) )
5 expaddz 11162 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( A ^ (
y  +  z ) )  =  ( ( A ^ y )  x.  ( A ^
z ) ) )
6 zaddcl 10075 . . . . . 6  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  +  z )  e.  ZZ )
76adantl 452 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( y  +  z )  e.  ZZ )
8 oveq2 5882 . . . . . 6  |-  ( x  =  ( y  +  z )  ->  ( A ^ x )  =  ( A ^ (
y  +  z ) ) )
9 ovex 5899 . . . . . 6  |-  ( A ^ ( y  +  z ) )  e. 
_V
108, 3, 9fvmpt 5618 . . . . 5  |-  ( ( y  +  z )  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  ( y  +  z ) )  =  ( A ^
( y  +  z ) ) )
117, 10syl 15 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( ( x  e.  ZZ  |->  ( A ^
x ) ) `  ( y  +  z ) )  =  ( A ^ ( y  +  z ) ) )
12 oveq2 5882 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
13 ovex 5899 . . . . . . 7  |-  ( A ^ y )  e. 
_V
1412, 3, 13fvmpt 5618 . . . . . 6  |-  ( y  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  y )  =  ( A ^
y ) )
15 oveq2 5882 . . . . . . 7  |-  ( x  =  z  ->  ( A ^ x )  =  ( A ^ z
) )
16 ovex 5899 . . . . . . 7  |-  ( A ^ z )  e. 
_V
1715, 3, 16fvmpt 5618 . . . . . 6  |-  ( z  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  z )  =  ( A ^
z ) )
1814, 17oveqan12d 5893 . . . . 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 452 . . . 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 2338 . . 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 2648 . 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 16441 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
23 subrgsubg 15567 . . . . . 6  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
2422, 23ax-mp 8 . . . . 5  |-  ZZ  e.  (SubGrp ` fld )
25 expghm.1 . . . . . 6  |-  Z  =  (flds  ZZ )
2625subggrp 14640 . . . . 5  |-  ( ZZ  e.  (SubGrp ` fld )  ->  Z  e. 
Grp )
2724, 26ax-mp 8 . . . 4  |-  Z  e. 
Grp
28 cnrng 16412 . . . . 5  |-fld  e.  Ring
29 cnfldbas 16399 . . . . . . 7  |-  CC  =  ( Base ` fld )
30 cnfld0 16414 . . . . . . 7  |-  0  =  ( 0g ` fld )
31 cndrng 16419 . . . . . . 7  |-fld  e.  DivRing
3229, 30, 31drngui 15534 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
33 expghm.3 . . . . . . 7  |-  U  =  ( Ms  ( CC  \  { 0 } ) )
34 expghm.2 . . . . . . . 8  |-  M  =  (mulGrp ` fld )
3534oveq1i 5884 . . . . . . 7  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3633, 35eqtri 2316 . . . . . 6  |-  U  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3732, 36unitgrp 15465 . . . . 5  |-  (fld  e.  Ring  ->  U  e.  Grp )
3828, 37ax-mp 8 . . . 4  |-  U  e. 
Grp
3927, 38pm3.2i 441 . . 3  |-  ( Z  e.  Grp  /\  U  e.  Grp )
4025subrgbas 15570 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
4122, 40ax-mp 8 . . . 4  |-  ZZ  =  ( Base `  Z )
42 difss 3316 . . . . 5  |-  ( CC 
\  { 0 } )  C_  CC
4334, 29mgpbas 15347 . . . . . 6  |-  CC  =  ( Base `  M )
4433, 43ressbas2 13215 . . . . 5  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( CC  \  {
0 } )  =  ( Base `  U
) )
4542, 44ax-mp 8 . . . 4  |-  ( CC 
\  { 0 } )  =  ( Base `  U )
46 cnfldadd 16400 . . . . . 6  |-  +  =  ( +g  ` fld )
4725, 46ressplusg 13266 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
4822, 47ax-mp 8 . . . 4  |-  +  =  ( +g  `  Z )
49 fvex 5555 . . . . . 6  |-  (Unit ` fld )  e.  _V
5032, 49eqeltri 2366 . . . . 5  |-  ( CC 
\  { 0 } )  e.  _V
51 cnfldmul 16401 . . . . . . 7  |-  x.  =  ( .r ` fld )
5234, 51mgpplusg 15345 . . . . . 6  |-  x.  =  ( +g  `  M )
5333, 52ressplusg 13266 . . . . 5  |-  ( ( CC  \  { 0 } )  e.  _V  ->  x.  =  ( +g  `  U ) )
5450, 53ax-mp 8 . . . 4  |-  x.  =  ( +g  `  U )
5541, 45, 48, 54isghm 14699 . . 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 884 . 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 645 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( x  e.  ZZ  |->  ( A ^ x ) )  e.  ( Z 
GrpHom  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756    x. cmul 8758   ZZcz 10040   ^cexp 11120   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   Grpcgrp 14378  SubGrpcsubg 14631    GrpHom cghm 14696  mulGrpcmgp 15341   Ringcrg 15353  Unitcui 15437  SubRingcsubrg 15557  ℂfldccnfld 16393
This theorem is referenced by:  lgseisenlem4  20607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-seq 11063  df-exp 11121  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-ghm 14697  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-subrg 15559  df-cnfld 16394
  Copyright terms: Public domain W3C validator