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Theorem efexp 11033
Description: The exponential of 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 8816 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
2 mulcom 7532 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
31, 2sylan2 281 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
43fveq2d 5322 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( exp `  ( N  x.  A
) ) )
5 oveq2 5674 . . . . . 6  |-  ( j  =  0  ->  ( A  x.  j )  =  ( A  x.  0 ) )
65fveq2d 5322 . . . . 5  |-  ( j  =  0  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  0 ) ) )
7 oveq2 5674 . . . . 5  |-  ( j  =  0  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ 0 ) )
86, 7eqeq12d 2103 . . . 4  |-  ( j  =  0  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) ) )
9 oveq2 5674 . . . . . 6  |-  ( j  =  k  ->  ( A  x.  j )  =  ( A  x.  k ) )
109fveq2d 5322 . . . . 5  |-  ( j  =  k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  k )
) )
11 oveq2 5674 . . . . 5  |-  ( j  =  k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ k
) )
1210, 11eqeq12d 2103 . . . 4  |-  ( j  =  k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  k
) )  =  ( ( exp `  A
) ^ k ) ) )
13 oveq2 5674 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A  x.  j )  =  ( A  x.  ( k  +  1 ) ) )
1413fveq2d 5322 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  ( k  +  1 ) ) ) )
15 oveq2 5674 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ (
k  +  1 ) ) )
1614, 15eqeq12d 2103 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  (
k  +  1 ) ) )  =  ( ( exp `  A
) ^ ( k  +  1 ) ) ) )
17 oveq2 5674 . . . . . 6  |-  ( j  =  -u k  ->  ( A  x.  j )  =  ( A  x.  -u k ) )
1817fveq2d 5322 . . . . 5  |-  ( j  =  -u k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  -u k ) ) )
19 oveq2 5674 . . . . 5  |-  ( j  =  -u k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ -u k
) )
2018, 19eqeq12d 2103 . . . 4  |-  ( j  =  -u k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  -u k
) )  =  ( ( exp `  A
) ^ -u k
) ) )
21 oveq2 5674 . . . . . 6  |-  ( j  =  N  ->  ( A  x.  j )  =  ( A  x.  N ) )
2221fveq2d 5322 . . . . 5  |-  ( j  =  N  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  N )
) )
23 oveq2 5674 . . . . 5  |-  ( j  =  N  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ N
) )
2422, 23eqeq12d 2103 . . . 4  |-  ( j  =  N  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  N
) )  =  ( ( exp `  A
) ^ N ) ) )
25 ef0 11023 . . . . 5  |-  ( exp `  0 )  =  1
26 mul01 7928 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
2726fveq2d 5322 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( exp `  0
) )
28 efcl 11015 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2928exp0d 10141 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  A
) ^ 0 )  =  1 )
3025, 27, 293eqtr4a 2147 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) )
31 oveq1 5673 . . . . . . 7  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) )  =  ( ( ( exp `  A
) ^ k )  x.  ( exp `  A
) ) )
3231adantl 272 . . . . . 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 8744 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  k  e.  CC )
34 ax-1cn 7499 . . . . . . . . . . . 12  |-  1  e.  CC
35 adddi 7535 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( A  x.  ( k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
3634, 35mp3an3 1263 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
37 mulid1 7546 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
3837adantr 271 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  1 )  =  A )
3938oveq2d 5682 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( ( A  x.  k )  +  ( A  x.  1 ) )  =  ( ( A  x.  k )  +  A ) )
4036, 39eqtrd 2121 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4133, 40sylan2 281 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4241fveq2d 5322 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( exp `  ( ( A  x.  k )  +  A
) ) )
43 mulcl 7530 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  k
)  e.  CC )
4433, 43sylan2 281 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  k
)  e.  CC )
45 simpl 108 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
46 efadd 11026 . . . . . . . . 9  |-  ( ( ( A  x.  k
)  e.  CC  /\  A  e.  CC )  ->  ( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4744, 45, 46syl2anc 404 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4842, 47eqtrd 2121 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4948adantr 271 . . . . . 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 10023 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5128, 50sylan 278 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5251adantr 271 . . . . . 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 2131 . . . . 5  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k ) )  ->  ( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  A ) ^ ( k  +  1 ) ) )
5453exp31 357 . . . 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 5674 . . . . . 6  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( 1  /  ( exp `  ( A  x.  k )
) )  =  ( 1  /  ( ( exp `  A ) ^ k ) ) )
56 nncn 8491 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  CC )
57 mulneg2 7935 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5856, 57sylan2 281 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5958fveq2d 5322 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( exp `  -u ( A  x.  k ) ) )
6056, 43sylan2 281 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  k
)  e.  CC )
61 efneg 11030 . . . . . . . . 9  |-  ( ( A  x.  k )  e.  CC  ->  ( exp `  -u ( A  x.  k ) )  =  ( 1  /  ( exp `  ( A  x.  k ) ) ) )
6260, 61syl 14 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  -u ( A  x.  k )
)  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
6359, 62eqtrd 2121 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
64 efap0 11028 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  A ) #  0 )
65 nnnn0 8741 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
66 expnegap0 10024 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  ( exp `  A ) #  0  /\  k  e. 
NN0 )  ->  (
( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6728, 64, 65, 66syl2an3an 1235 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6863, 67eqeq12d 2103 . . . . . 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 ) ) ) )
6955, 68syl5ibr 155 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k )  -> 
( exp `  ( A  x.  -u k ) )  =  ( ( exp `  A ) ^ -u k ) ) )
7069ex 114 . . . 4  |-  ( A  e.  CC  ->  (
k  e.  NN  ->  ( ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k )  -> 
( exp `  ( A  x.  -u k ) )  =  ( ( exp `  A ) ^ -u k ) ) ) )
718, 12, 16, 20, 24, 30, 54, 70zindd 8925 . . 3  |-  ( A  e.  CC  ->  ( N  e.  ZZ  ->  ( exp `  ( A  x.  N ) )  =  ( ( exp `  A ) ^ N
) ) )
7271imp 123 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( ( exp `  A ) ^ N ) )
734, 72eqtr3d 2123 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 103    = wceq 1290    e. wcel 1439   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   CCcc 7409   0cc0 7411   1c1 7412    + caddc 7414    x. cmul 7416   -ucneg 7715   # cap 8119    / cdiv 8200   NNcn 8483   NN0cn0 8734   ZZcz 8811   ^cexp 10015   expce 10993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-disj 3829  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-sup 6733  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-ico 9373  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-fac 10195  df-bc 10217  df-ihash 10245  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804  df-ef 10999
This theorem is referenced by:  efzval  11034  efgt0  11035  tanval3ap  11066  demoivre  11123
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