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Theorem expmhm 17989
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 11984 . . 3  |-  ( ( A  e.  CC  /\  x  e.  NN0 )  -> 
( A ^ x
)  e.  CC )
2 eqid 2451 . . 3  |-  ( x  e.  NN0  |->  ( A ^ x ) )  =  ( x  e. 
NN0  |->  ( A ^
x ) )
31, 2fmptd 5966 . 2  |-  ( A  e.  CC  ->  (
x  e.  NN0  |->  ( A ^ x ) ) : NN0 --> CC )
4 expadd 12007 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  NN0  /\  z  e.  NN0 )  ->  ( A ^ ( y  +  z ) )  =  ( ( A ^
y )  x.  ( A ^ z ) ) )
543expb 1189 . . . 4  |-  ( ( A  e.  CC  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( A ^ ( y  +  z ) )  =  ( ( A ^
y )  x.  ( A ^ z ) ) )
6 nn0addcl 10716 . . . . . 6  |-  ( ( y  e.  NN0  /\  z  e.  NN0 )  -> 
( y  +  z )  e.  NN0 )
76adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( y  +  z )  e. 
NN0 )
8 oveq2 6198 . . . . . 6  |-  ( x  =  ( y  +  z )  ->  ( A ^ x )  =  ( A ^ (
y  +  z ) ) )
9 ovex 6215 . . . . . 6  |-  ( A ^ ( y  +  z ) )  e. 
_V
108, 2, 9fvmpt 5873 . . . . 5  |-  ( ( y  +  z )  e.  NN0  ->  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( A ^ (
y  +  z ) ) )
117, 10syl 16 . . . 4  |-  ( ( A  e.  CC  /\  ( y  e.  NN0  /\  z  e.  NN0 )
)  ->  ( (
x  e.  NN0  |->  ( A ^ x ) ) `
 ( y  +  z ) )  =  ( A ^ (
y  +  z ) ) )
12 oveq2 6198 . . . . . . 7  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
13 ovex 6215 . . . . . . 7  |-  ( A ^ y )  e. 
_V
1412, 2, 13fvmpt 5873 . . . . . 6  |-  ( y  e.  NN0  ->  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 y )  =  ( A ^ y
) )
15 oveq2 6198 . . . . . . 7  |-  ( x  =  z  ->  ( A ^ x )  =  ( A ^ z
) )
16 ovex 6215 . . . . . . 7  |-  ( A ^ z )  e. 
_V
1715, 2, 16fvmpt 5873 . . . . . 6  |-  ( z  e.  NN0  ->  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 z )  =  ( A ^ z
) )
1814, 17oveqan12d 6209 . . . . 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 466 . . . 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 2502 . . 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 2904 . 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 10695 . . . 4  |-  0  e.  NN0
23 oveq2 6198 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
24 ovex 6215 . . . . 5  |-  ( A ^ 0 )  e. 
_V
2523, 2, 24fvmpt 5873 . . . 4  |-  ( 0  e.  NN0  ->  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 0 )  =  ( A ^ 0 ) )
2622, 25ax-mp 5 . . 3  |-  ( ( x  e.  NN0  |->  ( A ^ x ) ) `
 0 )  =  ( A ^ 0 )
27 exp0 11970 . . 3  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2826, 27syl5eq 2504 . 2  |-  ( A  e.  CC  ->  (
( x  e.  NN0  |->  ( A ^ x ) ) `  0 )  =  1 )
29 nn0subm 17977 . . . . 5  |-  NN0  e.  (SubMnd ` fld )
30 expmhm.1 . . . . . 6  |-  N  =  (flds  NN0 )
3130submmnd 15584 . . . . 5  |-  ( NN0 
e.  (SubMnd ` fld )  ->  N  e. 
Mnd )
3229, 31ax-mp 5 . . . 4  |-  N  e. 
Mnd
33 cnrng 17947 . . . . 5  |-fld  e.  Ring
34 expmhm.2 . . . . . 6  |-  M  =  (mulGrp ` fld )
3534rngmgp 16757 . . . . 5  |-  (fld  e.  Ring  ->  M  e.  Mnd )
3633, 35ax-mp 5 . . . 4  |-  M  e. 
Mnd
3732, 36pm3.2i 455 . . 3  |-  ( N  e.  Mnd  /\  M  e.  Mnd )
3830submbas 15585 . . . . 5  |-  ( NN0 
e.  (SubMnd ` fld )  ->  NN0  =  ( Base `  N )
)
3929, 38ax-mp 5 . . . 4  |-  NN0  =  ( Base `  N )
40 cnfldbas 17931 . . . . 5  |-  CC  =  ( Base ` fld )
4134, 40mgpbas 16702 . . . 4  |-  CC  =  ( Base `  M )
42 cnfldadd 17932 . . . . . 6  |-  +  =  ( +g  ` fld )
4330, 42ressplusg 14382 . . . . 5  |-  ( NN0 
e.  (SubMnd ` fld )  ->  +  =  ( +g  `  N ) )
4429, 43ax-mp 5 . . . 4  |-  +  =  ( +g  `  N )
45 cnfldmul 17933 . . . . 5  |-  x.  =  ( .r ` fld )
4634, 45mgpplusg 16700 . . . 4  |-  x.  =  ( +g  `  M )
47 cnfld0 17949 . . . . . 6  |-  0  =  ( 0g ` fld )
4830, 47subm0 15586 . . . . 5  |-  ( NN0 
e.  (SubMnd ` fld )  ->  0  =  ( 0g `  N
) )
4929, 48ax-mp 5 . . . 4  |-  0  =  ( 0g `  N )
50 cnfld1 17950 . . . . 5  |-  1  =  ( 1r ` fld )
5134, 50rngidval 16710 . . . 4  |-  1  =  ( 0g `  M )
5239, 41, 44, 46, 49, 51ismhm 15568 . . 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 909 . 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 1172 1  |-  ( A  e.  CC  ->  (
x  e.  NN0  |->  ( A ^ x ) )  e.  ( N MndHom  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795    |-> cmpt 4448   -->wf 5512   ` cfv 5516  (class class class)co 6190   CCcc 9381   0cc0 9383   1c1 9384    + caddc 9386    x. cmul 9388   NN0cn0 10680   ^cexp 11966   Basecbs 14276   ↾s cress 14277   +g cplusg 14340   0gc0g 14480   Mndcmnd 15511   MndHom cmhm 15564  SubMndcsubmnd 15565  mulGrpcmgp 16696   Ringcrg 16751  ℂfldccnfld 17927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-seq 11908  df-exp 11967  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-0g 14482  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-cmn 16383  df-mgp 16697  df-ur 16709  df-rng 16753  df-cring 16754  df-cnfld 17928
This theorem is referenced by: (None)
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