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Theorem expghm 16779
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 11407 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  ( CC  \  {
0 } ) )
213expa 1154 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  ( CC  \  { 0 } ) )
3 eqid 2438 . . 3  |-  ( x  e.  ZZ  |->  ( A ^ x ) )  =  ( x  e.  ZZ  |->  ( A ^
x ) )
42, 3fmptd 5895 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( x  e.  ZZ  |->  ( A ^ x ) ) : ZZ --> ( CC 
\  { 0 } ) )
5 expaddz 11426 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( A ^ (
y  +  z ) )  =  ( ( A ^ y )  x.  ( A ^
z ) ) )
6 zaddcl 10319 . . . . . 6  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  +  z )  e.  ZZ )
76adantl 454 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( y  +  z )  e.  ZZ )
8 oveq2 6091 . . . . . 6  |-  ( x  =  ( y  +  z )  ->  ( A ^ x )  =  ( A ^ (
y  +  z ) ) )
9 ovex 6108 . . . . . 6  |-  ( A ^ ( y  +  z ) )  e. 
_V
108, 3, 9fvmpt 5808 . . . . 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 6091 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
13 ovex 6108 . . . . . . 7  |-  ( A ^ y )  e. 
_V
1412, 3, 13fvmpt 5808 . . . . . 6  |-  ( y  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  y )  =  ( A ^
y ) )
15 oveq2 6091 . . . . . . 7  |-  ( x  =  z  ->  ( A ^ x )  =  ( A ^ z
) )
16 ovex 6108 . . . . . . 7  |-  ( A ^ z )  e. 
_V
1715, 3, 16fvmpt 5808 . . . . . 6  |-  ( z  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  z )  =  ( A ^
z ) )
1814, 17oveqan12d 6102 . . . . 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 454 . . . 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 2480 . . 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 2800 . 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 16754 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
23 subrgsubg 15876 . . . . . 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 14949 . . . . 5  |-  ( ZZ  e.  (SubGrp ` fld )  ->  Z  e. 
Grp )
2724, 26ax-mp 8 . . . 4  |-  Z  e. 
Grp
28 cnrng 16725 . . . . 5  |-fld  e.  Ring
29 cnfldbas 16709 . . . . . . 7  |-  CC  =  ( Base ` fld )
30 cnfld0 16727 . . . . . . 7  |-  0  =  ( 0g ` fld )
31 cndrng 16732 . . . . . . 7  |-fld  e.  DivRing
3229, 30, 31drngui 15843 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
33 expghm.3 . . . . . . 7  |-  U  =  ( Ms  ( CC  \  { 0 } ) )
34 expghm.2 . . . . . . . 8  |-  M  =  (mulGrp ` fld )
3534oveq1i 6093 . . . . . . 7  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3633, 35eqtri 2458 . . . . . 6  |-  U  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3732, 36unitgrp 15774 . . . . 5  |-  (fld  e.  Ring  ->  U  e.  Grp )
3828, 37ax-mp 8 . . . 4  |-  U  e. 
Grp
3927, 38pm3.2i 443 . . 3  |-  ( Z  e.  Grp  /\  U  e.  Grp )
4025subrgbas 15879 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
4122, 40ax-mp 8 . . . 4  |-  ZZ  =  ( Base `  Z )
42 difss 3476 . . . . 5  |-  ( CC 
\  { 0 } )  C_  CC
4334, 29mgpbas 15656 . . . . . 6  |-  CC  =  ( Base `  M )
4433, 43ressbas2 13522 . . . . 5  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( CC  \  {
0 } )  =  ( Base `  U
) )
4542, 44ax-mp 8 . . . 4  |-  ( CC 
\  { 0 } )  =  ( Base `  U )
46 cnfldadd 16710 . . . . . 6  |-  +  =  ( +g  ` fld )
4725, 46ressplusg 13573 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
4822, 47ax-mp 8 . . . 4  |-  +  =  ( +g  `  Z )
49 fvex 5744 . . . . . 6  |-  (Unit ` fld )  e.  _V
5032, 49eqeltri 2508 . . . . 5  |-  ( CC 
\  { 0 } )  e.  _V
51 cnfldmul 16711 . . . . . . 7  |-  x.  =  ( .r ` fld )
5234, 51mgpplusg 15654 . . . . . 6  |-  x.  =  ( +g  `  M )
5333, 52ressplusg 13573 . . . . 5  |-  ( ( CC  \  { 0 } )  e.  _V  ->  x.  =  ( +g  `  U ) )
5450, 53ax-mp 8 . . . 4  |-  x.  =  ( +g  `  U )
5541, 45, 48, 54isghm 15008 . . 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 886 . 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 647 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 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958    \ cdif 3319    C_ wss 3322   {csn 3816    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    + caddc 8995    x. cmul 8997   ZZcz 10284   ^cexp 11384   Basecbs 13471   ↾s cress 13472   +g cplusg 13531   Grpcgrp 14687  SubGrpcsubg 14940    GrpHom cghm 15005  mulGrpcmgp 15650   Ringcrg 15662  Unitcui 15746  SubRingcsubrg 15866  ℂfldccnfld 16705
This theorem is referenced by:  lgseisenlem4  21138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-seq 11326  df-exp 11385  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-subg 14943  df-ghm 15006  df-cmn 15416  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-drng 15839  df-subrg 15868  df-cnfld 16706
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