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Theorem efexp 12329
Description: Exponential function to an integer power. Corollary 15-4.4 of [Gleason] p. 309, restricted to integers. (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
Assertion
Ref Expression
efexp  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( N  x.  A )
)  =  ( ( exp `  A ) ^ N ) )

Proof of Theorem efexp
StepHypRef Expression
1 zcn 9982 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
2 mulcom 8777 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
31, 2sylan2 462 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
43fveq2d 5448 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( exp `  ( N  x.  A
) ) )
5 oveq2 5786 . . . . . 6  |-  ( j  =  0  ->  ( A  x.  j )  =  ( A  x.  0 ) )
65fveq2d 5448 . . . . 5  |-  ( j  =  0  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  0 ) ) )
7 oveq2 5786 . . . . 5  |-  ( j  =  0  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ 0 ) )
86, 7eqeq12d 2270 . . . 4  |-  ( j  =  0  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) ) )
9 oveq2 5786 . . . . . 6  |-  ( j  =  k  ->  ( A  x.  j )  =  ( A  x.  k ) )
109fveq2d 5448 . . . . 5  |-  ( j  =  k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  k )
) )
11 oveq2 5786 . . . . 5  |-  ( j  =  k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ k
) )
1210, 11eqeq12d 2270 . . . 4  |-  ( j  =  k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  k
) )  =  ( ( exp `  A
) ^ k ) ) )
13 oveq2 5786 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A  x.  j )  =  ( A  x.  ( k  +  1 ) ) )
1413fveq2d 5448 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  ( k  +  1 ) ) ) )
15 oveq2 5786 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ (
k  +  1 ) ) )
1614, 15eqeq12d 2270 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  (
k  +  1 ) ) )  =  ( ( exp `  A
) ^ ( k  +  1 ) ) ) )
17 oveq2 5786 . . . . . 6  |-  ( j  =  -u k  ->  ( A  x.  j )  =  ( A  x.  -u k ) )
1817fveq2d 5448 . . . . 5  |-  ( j  =  -u k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  -u k ) ) )
19 oveq2 5786 . . . . 5  |-  ( j  =  -u k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ -u k
) )
2018, 19eqeq12d 2270 . . . 4  |-  ( j  =  -u k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  -u k
) )  =  ( ( exp `  A
) ^ -u k
) ) )
21 oveq2 5786 . . . . . 6  |-  ( j  =  N  ->  ( A  x.  j )  =  ( A  x.  N ) )
2221fveq2d 5448 . . . . 5  |-  ( j  =  N  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  N )
) )
23 oveq2 5786 . . . . 5  |-  ( j  =  N  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ N
) )
2422, 23eqeq12d 2270 . . . 4  |-  ( j  =  N  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  N
) )  =  ( ( exp `  A
) ^ N ) ) )
25 ef0 12320 . . . . 5  |-  ( exp `  0 )  =  1
26 mul01 8945 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
2726fveq2d 5448 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( exp `  0
) )
28 efcl 12312 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2928exp0d 11191 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  A
) ^ 0 )  =  1 )
3025, 27, 293eqtr4a 2314 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) )
31 oveq1 5785 . . . . . . 7  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) )  =  ( ( ( exp `  A
) ^ k )  x.  ( exp `  A
) ) )
3231adantl 454 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k ) )  ->  ( ( exp `  ( A  x.  k
) )  x.  ( exp `  A ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
33 nn0cn 9928 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  k  e.  CC )
34 ax-1cn 8749 . . . . . . . . . . . 12  |-  1  e.  CC
35 adddi 8780 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( A  x.  ( k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
3634, 35mp3an3 1271 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
37 mulid1 8788 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
3837adantr 453 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  1 )  =  A )
3938oveq2d 5794 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( ( A  x.  k )  +  ( A  x.  1 ) )  =  ( ( A  x.  k )  +  A ) )
4036, 39eqtrd 2288 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4133, 40sylan2 462 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4241fveq2d 5448 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( exp `  ( ( A  x.  k )  +  A
) ) )
43 mulcl 8775 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  k
)  e.  CC )
4433, 43sylan2 462 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  k
)  e.  CC )
45 simpl 445 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
46 efadd 12323 . . . . . . . . 9  |-  ( ( ( A  x.  k
)  e.  CC  /\  A  e.  CC )  ->  ( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4744, 45, 46syl2anc 645 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4842, 47eqtrd 2288 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4948adantr 453 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k ) )  ->  ( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
50 expp1 11062 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5128, 50sylan 459 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5251adantr 453 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k ) )  ->  ( ( exp `  A ) ^ (
k  +  1 ) )  =  ( ( ( exp `  A
) ^ k )  x.  ( exp `  A
) ) )
5332, 49, 523eqtr4d 2298 . . . . 5  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k ) )  ->  ( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  A ) ^ ( k  +  1 ) ) )
5453exp31 590 . . . 4  |-  ( A  e.  CC  ->  (
k  e.  NN0  ->  ( ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  A ) ^ ( k  +  1 ) ) ) ) )
55 oveq2 5786 . . . . . 6  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( 1  /  ( exp `  ( A  x.  k )
) )  =  ( 1  /  ( ( exp `  A ) ^ k ) ) )
56 nncn 9708 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  CC )
57 mulneg2 9171 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5856, 57sylan2 462 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5958fveq2d 5448 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( exp `  -u ( A  x.  k ) ) )
6056, 43sylan2 462 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  k
)  e.  CC )
61 efneg 12326 . . . . . . . . 9  |-  ( ( A  x.  k )  e.  CC  ->  ( exp `  -u ( A  x.  k ) )  =  ( 1  /  ( exp `  ( A  x.  k ) ) ) )
6260, 61syl 17 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  -u ( A  x.  k )
)  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
6359, 62eqtrd 2288 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
64 nnnn0 9925 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
65 expneg 11063 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6628, 64, 65syl2an 465 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6763, 66eqeq12d 2270 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  ( A  x.  -u k ) )  =  ( ( exp `  A ) ^ -u k )  <-> 
( 1  /  ( exp `  ( A  x.  k ) ) )  =  ( 1  / 
( ( exp `  A
) ^ k ) ) ) )
6855, 67syl5ibr 214 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k )  -> 
( exp `  ( A  x.  -u k ) )  =  ( ( exp `  A ) ^ -u k ) ) )
6968ex 425 . . . 4  |-  ( A  e.  CC  ->  (
k  e.  NN  ->  ( ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k )  -> 
( exp `  ( A  x.  -u k ) )  =  ( ( exp `  A ) ^ -u k ) ) ) )
708, 12, 16, 20, 24, 30, 54, 69zindd 10066 . . 3  |-  ( A  e.  CC  ->  ( N  e.  ZZ  ->  ( exp `  ( A  x.  N ) )  =  ( ( exp `  A ) ^ N
) ) )
7170imp 420 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( ( exp `  A ) ^ N ) )
724, 71eqtr3d 2290 1  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( N  x.  A )
)  =  ( ( exp `  A ) ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   ` cfv 4659  (class class class)co 5778   CCcc 8689   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696   -ucneg 8992    / cdiv 9377   NNcn 9700   NN0cn0 9918   ZZcz 9977   ^cexp 11056   expce 12291
This theorem is referenced by:  efzval  12330  efgt0  12331  tanval3  12362  demoivre  12428  ef2kpi  19794  efif1olem4  19855  explog  19895  reexplog  19896  relogexp  19897  tanarg  19918  root1eq1  20043
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-pm 6729  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-oi 7179  df-card 7526  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-ico 10614  df-fz 10735  df-fzo 10823  df-fl 10877  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297
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