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Theorem efexp 12397
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
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zcn 10045 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
2 mulcom 8839 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
31, 2sylan2 460 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
43fveq2d 5545 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( exp `  ( N  x.  A
) ) )
5 oveq2 5882 . . . . . 6  |-  ( j  =  0  ->  ( A  x.  j )  =  ( A  x.  0 ) )
65fveq2d 5545 . . . . 5  |-  ( j  =  0  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  0 ) ) )
7 oveq2 5882 . . . . 5  |-  ( j  =  0  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ 0 ) )
86, 7eqeq12d 2310 . . . 4  |-  ( j  =  0  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) ) )
9 oveq2 5882 . . . . . 6  |-  ( j  =  k  ->  ( A  x.  j )  =  ( A  x.  k ) )
109fveq2d 5545 . . . . 5  |-  ( j  =  k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  k )
) )
11 oveq2 5882 . . . . 5  |-  ( j  =  k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ k
) )
1210, 11eqeq12d 2310 . . . 4  |-  ( j  =  k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  k
) )  =  ( ( exp `  A
) ^ k ) ) )
13 oveq2 5882 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A  x.  j )  =  ( A  x.  ( k  +  1 ) ) )
1413fveq2d 5545 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  ( k  +  1 ) ) ) )
15 oveq2 5882 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ (
k  +  1 ) ) )
1614, 15eqeq12d 2310 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  (
k  +  1 ) ) )  =  ( ( exp `  A
) ^ ( k  +  1 ) ) ) )
17 oveq2 5882 . . . . . 6  |-  ( j  =  -u k  ->  ( A  x.  j )  =  ( A  x.  -u k ) )
1817fveq2d 5545 . . . . 5  |-  ( j  =  -u k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  -u k ) ) )
19 oveq2 5882 . . . . 5  |-  ( j  =  -u k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ -u k
) )
2018, 19eqeq12d 2310 . . . 4  |-  ( j  =  -u k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  -u k
) )  =  ( ( exp `  A
) ^ -u k
) ) )
21 oveq2 5882 . . . . . 6  |-  ( j  =  N  ->  ( A  x.  j )  =  ( A  x.  N ) )
2221fveq2d 5545 . . . . 5  |-  ( j  =  N  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  N )
) )
23 oveq2 5882 . . . . 5  |-  ( j  =  N  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ N
) )
2422, 23eqeq12d 2310 . . . 4  |-  ( j  =  N  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  N
) )  =  ( ( exp `  A
) ^ N ) ) )
25 ef0 12388 . . . . 5  |-  ( exp `  0 )  =  1
26 mul01 9007 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
2726fveq2d 5545 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( exp `  0
) )
28 efcl 12380 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2928exp0d 11255 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  A
) ^ 0 )  =  1 )
3025, 27, 293eqtr4a 2354 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) )
31 oveq1 5881 . . . . . . 7  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) )  =  ( ( ( exp `  A
) ^ k )  x.  ( exp `  A
) ) )
3231adantl 452 . . . . . 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 9991 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  k  e.  CC )
34 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
35 adddi 8842 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( A  x.  ( k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
3634, 35mp3an3 1266 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
37 mulid1 8851 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
3837adantr 451 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  1 )  =  A )
3938oveq2d 5890 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( ( A  x.  k )  +  ( A  x.  1 ) )  =  ( ( A  x.  k )  +  A ) )
4036, 39eqtrd 2328 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4133, 40sylan2 460 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4241fveq2d 5545 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( exp `  ( ( A  x.  k )  +  A
) ) )
43 mulcl 8837 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  k
)  e.  CC )
4433, 43sylan2 460 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  k
)  e.  CC )
45 simpl 443 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
46 efadd 12391 . . . . . . . . 9  |-  ( ( ( A  x.  k
)  e.  CC  /\  A  e.  CC )  ->  ( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4744, 45, 46syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4842, 47eqtrd 2328 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4948adantr 451 . . . . . 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 11126 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5128, 50sylan 457 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5251adantr 451 . . . . . 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 2338 . . . . 5  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k ) )  ->  ( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  A ) ^ ( k  +  1 ) ) )
5453exp31 587 . . . 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 5882 . . . . . 6  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( 1  /  ( exp `  ( A  x.  k )
) )  =  ( 1  /  ( ( exp `  A ) ^ k ) ) )
56 nncn 9770 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  CC )
57 mulneg2 9233 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5856, 57sylan2 460 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5958fveq2d 5545 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( exp `  -u ( A  x.  k ) ) )
6056, 43sylan2 460 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  k
)  e.  CC )
61 efneg 12394 . . . . . . . . 9  |-  ( ( A  x.  k )  e.  CC  ->  ( exp `  -u ( A  x.  k ) )  =  ( 1  /  ( exp `  ( A  x.  k ) ) ) )
6260, 61syl 15 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  -u ( A  x.  k )
)  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
6359, 62eqtrd 2328 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
64 nnnn0 9988 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
65 expneg 11127 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6628, 64, 65syl2an 463 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6763, 66eqeq12d 2310 . . . . . 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 212 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k )  -> 
( exp `  ( A  x.  -u k ) )  =  ( ( exp `  A ) ^ -u k ) ) )
6968ex 423 . . . 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 10129 . . 3  |-  ( A  e.  CC  ->  ( N  e.  ZZ  ->  ( exp `  ( A  x.  N ) )  =  ( ( exp `  A ) ^ N
) ) )
7170imp 418 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( ( exp `  A ) ^ N ) )
724, 71eqtr3d 2330 1  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( N  x.  A )
)  =  ( ( exp `  A ) ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   -ucneg 9054    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120   expce 12359
This theorem is referenced by:  efzval  12398  efgt0  12399  tanval3  12430  demoivre  12496  ef2kpi  19862  efif1olem4  19923  explog  19963  reexplog  19964  relogexp  19965  tanarg  19986  root1eq1  20111
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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-pre-sup 8831  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-se 4369  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-isom 5280  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-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365
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