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Theorem expghm 16732
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 11360 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  x  e.  ZZ )  ->  ( A ^ x )  e.  ( CC  \  {
0 } ) )
213expa 1153 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  x  e.  ZZ )  ->  ( A ^
x )  e.  ( CC  \  { 0 } ) )
3 eqid 2404 . . 3  |-  ( x  e.  ZZ  |->  ( A ^ x ) )  =  ( x  e.  ZZ  |->  ( A ^
x ) )
42, 3fmptd 5852 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( x  e.  ZZ  |->  ( A ^ x ) ) : ZZ --> ( CC 
\  { 0 } ) )
5 expaddz 11379 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( A ^ (
y  +  z ) )  =  ( ( A ^ y )  x.  ( A ^
z ) ) )
6 zaddcl 10273 . . . . . 6  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  +  z )  e.  ZZ )
76adantl 453 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( y  +  z )  e.  ZZ )
8 oveq2 6048 . . . . . 6  |-  ( x  =  ( y  +  z )  ->  ( A ^ x )  =  ( A ^ (
y  +  z ) ) )
9 ovex 6065 . . . . . 6  |-  ( A ^ ( y  +  z ) )  e. 
_V
108, 3, 9fvmpt 5765 . . . . 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 6048 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
13 ovex 6065 . . . . . . 7  |-  ( A ^ y )  e. 
_V
1412, 3, 13fvmpt 5765 . . . . . 6  |-  ( y  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  y )  =  ( A ^
y ) )
15 oveq2 6048 . . . . . . 7  |-  ( x  =  z  ->  ( A ^ x )  =  ( A ^ z
) )
16 ovex 6065 . . . . . . 7  |-  ( A ^ z )  e. 
_V
1715, 3, 16fvmpt 5765 . . . . . 6  |-  ( z  e.  ZZ  ->  (
( x  e.  ZZ  |->  ( A ^ x ) ) `  z )  =  ( A ^
z ) )
1814, 17oveqan12d 6059 . . . . 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 453 . . . 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 2446 . . 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 2758 . 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 16707 . . . . . 6  |-  ZZ  e.  (SubRing ` fld )
23 subrgsubg 15829 . . . . . 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 14902 . . . . 5  |-  ( ZZ  e.  (SubGrp ` fld )  ->  Z  e. 
Grp )
2724, 26ax-mp 8 . . . 4  |-  Z  e. 
Grp
28 cnrng 16678 . . . . 5  |-fld  e.  Ring
29 cnfldbas 16662 . . . . . . 7  |-  CC  =  ( Base ` fld )
30 cnfld0 16680 . . . . . . 7  |-  0  =  ( 0g ` fld )
31 cndrng 16685 . . . . . . 7  |-fld  e.  DivRing
3229, 30, 31drngui 15796 . . . . . 6  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
33 expghm.3 . . . . . . 7  |-  U  =  ( Ms  ( CC  \  { 0 } ) )
34 expghm.2 . . . . . . . 8  |-  M  =  (mulGrp ` fld )
3534oveq1i 6050 . . . . . . 7  |-  ( Ms  ( CC  \  { 0 } ) )  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3633, 35eqtri 2424 . . . . . 6  |-  U  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
3732, 36unitgrp 15727 . . . . 5  |-  (fld  e.  Ring  ->  U  e.  Grp )
3828, 37ax-mp 8 . . . 4  |-  U  e. 
Grp
3927, 38pm3.2i 442 . . 3  |-  ( Z  e.  Grp  /\  U  e.  Grp )
4025subrgbas 15832 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
4122, 40ax-mp 8 . . . 4  |-  ZZ  =  ( Base `  Z )
42 difss 3434 . . . . 5  |-  ( CC 
\  { 0 } )  C_  CC
4334, 29mgpbas 15609 . . . . . 6  |-  CC  =  ( Base `  M )
4433, 43ressbas2 13475 . . . . 5  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( CC  \  {
0 } )  =  ( Base `  U
) )
4542, 44ax-mp 8 . . . 4  |-  ( CC 
\  { 0 } )  =  ( Base `  U )
46 cnfldadd 16663 . . . . . 6  |-  +  =  ( +g  ` fld )
4725, 46ressplusg 13526 . . . . 5  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
4822, 47ax-mp 8 . . . 4  |-  +  =  ( +g  `  Z )
49 fvex 5701 . . . . . 6  |-  (Unit ` fld )  e.  _V
5032, 49eqeltri 2474 . . . . 5  |-  ( CC 
\  { 0 } )  e.  _V
51 cnfldmul 16664 . . . . . . 7  |-  x.  =  ( .r ` fld )
5234, 51mgpplusg 15607 . . . . . 6  |-  x.  =  ( +g  `  M )
5333, 52ressplusg 13526 . . . . 5  |-  ( ( CC  \  { 0 } )  e.  _V  ->  x.  =  ( +g  `  U ) )
5450, 53ax-mp 8 . . . 4  |-  x.  =  ( +g  `  U )
5541, 45, 48, 54isghm 14961 . . 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 885 . 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 646 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 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    + caddc 8949    x. cmul 8951   ZZcz 10238   ^cexp 11337   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   Grpcgrp 14640  SubGrpcsubg 14893    GrpHom cghm 14958  mulGrpcmgp 15603   Ringcrg 15615  Unitcui 15699  SubRingcsubrg 15819  ℂfldccnfld 16658
This theorem is referenced by:  lgseisenlem4  21089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-seq 11279  df-exp 11338  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-subg 14896  df-ghm 14959  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-subrg 15821  df-cnfld 16659
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