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Theorem efexp 11579
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 9172 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
2 mulcom 7861 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
31, 2sylan2 284 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
43fveq2d 5472 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( exp `  ( N  x.  A
) ) )
5 oveq2 5832 . . . . . 6  |-  ( j  =  0  ->  ( A  x.  j )  =  ( A  x.  0 ) )
65fveq2d 5472 . . . . 5  |-  ( j  =  0  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  0 ) ) )
7 oveq2 5832 . . . . 5  |-  ( j  =  0  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ 0 ) )
86, 7eqeq12d 2172 . . . 4  |-  ( j  =  0  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) ) )
9 oveq2 5832 . . . . . 6  |-  ( j  =  k  ->  ( A  x.  j )  =  ( A  x.  k ) )
109fveq2d 5472 . . . . 5  |-  ( j  =  k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  k )
) )
11 oveq2 5832 . . . . 5  |-  ( j  =  k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ k
) )
1210, 11eqeq12d 2172 . . . 4  |-  ( j  =  k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  k
) )  =  ( ( exp `  A
) ^ k ) ) )
13 oveq2 5832 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A  x.  j )  =  ( A  x.  ( k  +  1 ) ) )
1413fveq2d 5472 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  ( k  +  1 ) ) ) )
15 oveq2 5832 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ (
k  +  1 ) ) )
1614, 15eqeq12d 2172 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  (
k  +  1 ) ) )  =  ( ( exp `  A
) ^ ( k  +  1 ) ) ) )
17 oveq2 5832 . . . . . 6  |-  ( j  =  -u k  ->  ( A  x.  j )  =  ( A  x.  -u k ) )
1817fveq2d 5472 . . . . 5  |-  ( j  =  -u k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  -u k ) ) )
19 oveq2 5832 . . . . 5  |-  ( j  =  -u k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ -u k
) )
2018, 19eqeq12d 2172 . . . 4  |-  ( j  =  -u k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  -u k
) )  =  ( ( exp `  A
) ^ -u k
) ) )
21 oveq2 5832 . . . . . 6  |-  ( j  =  N  ->  ( A  x.  j )  =  ( A  x.  N ) )
2221fveq2d 5472 . . . . 5  |-  ( j  =  N  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  N )
) )
23 oveq2 5832 . . . . 5  |-  ( j  =  N  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ N
) )
2422, 23eqeq12d 2172 . . . 4  |-  ( j  =  N  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  N
) )  =  ( ( exp `  A
) ^ N ) ) )
25 ef0 11569 . . . . 5  |-  ( exp `  0 )  =  1
26 mul01 8264 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
2726fveq2d 5472 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( exp `  0
) )
28 efcl 11561 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2928exp0d 10545 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  A
) ^ 0 )  =  1 )
3025, 27, 293eqtr4a 2216 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) )
31 oveq1 5831 . . . . . . 7  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) )  =  ( ( ( exp `  A
) ^ k )  x.  ( exp `  A
) ) )
3231adantl 275 . . . . . 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 9100 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  k  e.  CC )
34 ax-1cn 7825 . . . . . . . . . . . 12  |-  1  e.  CC
35 adddi 7864 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( A  x.  ( k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
3634, 35mp3an3 1308 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
37 mulid1 7875 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
3837adantr 274 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  1 )  =  A )
3938oveq2d 5840 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( ( A  x.  k )  +  ( A  x.  1 ) )  =  ( ( A  x.  k )  +  A ) )
4036, 39eqtrd 2190 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4133, 40sylan2 284 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4241fveq2d 5472 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( exp `  ( ( A  x.  k )  +  A
) ) )
43 mulcl 7859 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  k
)  e.  CC )
4433, 43sylan2 284 . . . . . . . . 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 11572 . . . . . . . . 9  |-  ( ( ( A  x.  k
)  e.  CC  /\  A  e.  CC )  ->  ( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4744, 45, 46syl2anc 409 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4842, 47eqtrd 2190 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4948adantr 274 . . . . . 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 10426 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5128, 50sylan 281 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5251adantr 274 . . . . . 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 2200 . . . . 5  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k ) )  ->  ( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  A ) ^ ( k  +  1 ) ) )
5453exp31 362 . . . 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 5832 . . . . . 6  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( 1  /  ( exp `  ( A  x.  k )
) )  =  ( 1  /  ( ( exp `  A ) ^ k ) ) )
56 nncn 8841 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  CC )
57 mulneg2 8271 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5856, 57sylan2 284 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5958fveq2d 5472 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( exp `  -u ( A  x.  k ) ) )
6056, 43sylan2 284 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  k
)  e.  CC )
61 efneg 11576 . . . . . . . . 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 2190 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
64 efap0 11574 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  A ) #  0 )
65 nnnn0 9097 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
66 expnegap0 10427 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  ( exp `  A ) #  0  /\  k  e. 
NN0 )  ->  (
( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6728, 64, 65, 66syl2an3an 1280 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6863, 67eqeq12d 2172 . . . . . 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 9282 . . 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 2192 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 1335    e. wcel 2128   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   0cc0 7732   1c1 7733    + caddc 7735    x. cmul 7737   -ucneg 8047   # cap 8456    / cdiv 8545   NNcn 8833   NN0cn0 9090   ZZcz 9167   ^cexp 10418   expce 11539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-disj 3943  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-frec 6338  df-1o 6363  df-oadd 6367  df-er 6480  df-en 6686  df-dom 6687  df-fin 6688  df-sup 6928  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-ico 9798  df-fz 9913  df-fzo 10042  df-seqfrec 10345  df-exp 10419  df-fac 10600  df-bc 10622  df-ihash 10650  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176  df-sumdc 11251  df-ef 11545
This theorem is referenced by:  efzval  11580  efgt0  11581  tanval3ap  11611  demoivre  11669  ef2kpi  13138  reexplog  13203  relogexp  13204
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