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Theorem expmhm 16465
Description: Exponentiation is a monoid homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypotheses
Ref Expression
expmhm.1  |-  N  =  (flds  NN0 )
expmhm.2  |-  M  =  (mulGrp ` fld )
Assertion
Ref Expression
expmhm  |-  ( A  e.  CC  ->  (
x  e.  NN0  |->  ( A ^ x ) )  e.  ( N MndHom  M
) )
Distinct variable group:    x, A
Allowed substitution hints:    M( x)    N( x)

Proof of Theorem expmhm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 expcl 11137 . . 3  |-  ( ( A  e.  CC  /\  x  e.  NN0 )  -> 
( A ^ x
)  e.  CC )
2 eqid 2296 . . 3  |-  ( x  e.  NN0  |->  ( A ^ x ) )  =  ( x  e. 
NN0  |->  ( A ^
x ) )
31, 2fmptd 5700 . 2  |-  ( A  e.  CC  ->  (
x  e.  NN0  |->  ( A ^ x ) ) : NN0 --> CC )
4 expadd 11160 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  NN0  /\  z  e.  NN0 )  ->  ( A ^ ( y  +  z ) )  =  ( ( A ^
y )  x.  ( A ^ z ) ) )
543expb 1152 . . . 4  |-  ( ( A  e.  CC  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( A ^ ( y  +  z ) )  =  ( ( A ^
y )  x.  ( A ^ z ) ) )
6 nn0addcl 10015 . . . . . 6  |-  ( ( y  e.  NN0  /\  z  e.  NN0 )  -> 
( y  +  z )  e.  NN0 )
76adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( y  +  z )  e. 
NN0 )
8 oveq2 5882 . . . . . 6  |-  ( x  =  ( y  +  z )  ->  ( A ^ x )  =  ( A ^ (
y  +  z ) ) )
9 ovex 5899 . . . . . 6  |-  ( A ^ ( y  +  z ) )  e. 
_V
108, 2, 9fvmpt 5618 . . . . 5  |-  ( ( y  +  z )  e.  NN0  ->  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( A ^ (
y  +  z ) ) )
117, 10syl 15 . . . 4  |-  ( ( A  e.  CC  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( (
x  e.  NN0  |->  ( 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, 2, 13fvmpt 5618 . . . . . 6  |-  ( y  e.  NN0  ->  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 y )  =  ( A ^ y
) )
15 oveq2 5882 . . . . . . 7  |-  ( x  =  z  ->  ( A ^ x )  =  ( A ^ z
) )
16 ovex 5899 . . . . . . 7  |-  ( A ^ z )  e. 
_V
1715, 2, 16fvmpt 5618 . . . . . 6  |-  ( z  e.  NN0  ->  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 z )  =  ( A ^ z
) )
1814, 17oveqan12d 5893 . . . . 5  |-  ( ( y  e.  NN0  /\  z  e.  NN0 )  -> 
( ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e. 
NN0  |->  ( A ^
x ) ) `  z ) )  =  ( ( A ^
y )  x.  ( A ^ z ) ) )
1918adantl 452 . . . 4  |-  ( ( A  e.  CC  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( (
( x  e.  NN0  |->  ( A ^ x ) ) `  y )  x.  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 z ) )  =  ( ( A ^ y )  x.  ( A ^ z
) ) )
205, 11, 193eqtr4d 2338 . . 3  |-  ( ( A  e.  CC  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( (
x  e.  NN0  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e. 
NN0  |->  ( A ^
x ) ) `  z ) ) )
2120ralrimivva 2648 . 2  |-  ( A  e.  CC  ->  A. y  e.  NN0  A. z  e. 
NN0  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e. 
NN0  |->  ( A ^
x ) ) `  z ) ) )
22 0nn0 9996 . . . 4  |-  0  e.  NN0
23 oveq2 5882 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
24 ovex 5899 . . . . 5  |-  ( A ^ 0 )  e. 
_V
2523, 2, 24fvmpt 5618 . . . 4  |-  ( 0  e.  NN0  ->  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 0 )  =  ( A ^ 0 ) )
2622, 25ax-mp 8 . . 3  |-  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 0 )  =  ( A ^ 0 )
27 exp0 11124 . . 3  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2826, 27syl5eq 2340 . 2  |-  ( A  e.  CC  ->  (
( x  e.  NN0  |->  ( A ^ x ) ) `  0 )  =  1 )
29 nn0subm 16443 . . . . 5  |-  NN0  e.  (SubMnd ` fld )
30 expmhm.1 . . . . . 6  |-  N  =  (flds  NN0 )
3130submmnd 14447 . . . . 5  |-  ( NN0 
e.  (SubMnd ` fld )  ->  N  e. 
Mnd )
3229, 31ax-mp 8 . . . 4  |-  N  e. 
Mnd
33 cnrng 16412 . . . . 5  |-fld  e.  Ring
34 expmhm.2 . . . . . 6  |-  M  =  (mulGrp ` fld )
3534rngmgp 15363 . . . . 5  |-  (fld  e.  Ring  ->  M  e.  Mnd )
3633, 35ax-mp 8 . . . 4  |-  M  e. 
Mnd
3732, 36pm3.2i 441 . . 3  |-  ( N  e.  Mnd  /\  M  e.  Mnd )
3830submbas 14448 . . . . 5  |-  ( NN0 
e.  (SubMnd ` fld )  ->  NN0  =  ( Base `  N )
)
3929, 38ax-mp 8 . . . 4  |-  NN0  =  ( Base `  N )
40 cnfldbas 16399 . . . . 5  |-  CC  =  ( Base ` fld )
4134, 40mgpbas 15347 . . . 4  |-  CC  =  ( Base `  M )
42 cnfldadd 16400 . . . . . 6  |-  +  =  ( +g  ` fld )
4330, 42ressplusg 13266 . . . . 5  |-  ( NN0 
e.  (SubMnd ` fld )  ->  +  =  ( +g  `  N ) )
4429, 43ax-mp 8 . . . 4  |-  +  =  ( +g  `  N )
45 cnfldmul 16401 . . . . 5  |-  x.  =  ( .r ` fld )
4634, 45mgpplusg 15345 . . . 4  |-  x.  =  ( +g  `  M )
47 cnfld0 16414 . . . . . 6  |-  0  =  ( 0g ` fld )
4830, 47subm0 14449 . . . . 5  |-  ( NN0 
e.  (SubMnd ` fld )  ->  0  =  ( 0g `  N
) )
4929, 48ax-mp 8 . . . 4  |-  0  =  ( 0g `  N )
50 cnfld1 16415 . . . . 5  |-  1  =  ( 1r ` fld )
5134, 50rngidval 15359 . . . 4  |-  1  =  ( 0g `  M )
5239, 41, 44, 46, 49, 51ismhm 14433 . . 3  |-  ( ( x  e.  NN0  |->  ( A ^ x ) )  e.  ( N MndHom  M
)  <->  ( ( N  e.  Mnd  /\  M  e.  Mnd )  /\  (
( x  e.  NN0  |->  ( A ^ x ) ) : NN0 --> CC  /\  A. y  e.  NN0  A. z  e.  NN0  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e. 
NN0  |->  ( A ^
x ) ) `  z ) )  /\  ( ( x  e. 
NN0  |->  ( A ^
x ) ) ` 
0 )  =  1 ) ) )
5337, 52mpbiran 884 . 2  |-  ( ( x  e.  NN0  |->  ( A ^ x ) )  e.  ( N MndHom  M
)  <->  ( ( x  e.  NN0  |->  ( A ^ x ) ) : NN0 --> CC  /\  A. y  e.  NN0  A. z  e.  NN0  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 y )  x.  ( ( x  e. 
NN0  |->  ( A ^
x ) ) `  z ) )  /\  ( ( x  e. 
NN0  |->  ( A ^
x ) ) ` 
0 )  =  1 ) )
543, 21, 28, 53syl3anbrc 1136 1  |-  ( A  e.  CC  ->  (
x  e.  NN0  |->  ( A ^ x ) )  e.  ( N MndHom  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   NN0cn0 9981   ^cexp 11120   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   MndHom cmhm 14429  SubMndcsubmnd 14430  mulGrpcmgp 15341   Ringcrg 15353  ℂfldccnfld 16393
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-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-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  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-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-mhm 14431  df-submnd 14432  df-grp 14505  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-cnfld 16394
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