MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efexp Structured version   Unicode version

Theorem efexp 13848
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 10890 . . . 4  |-  ( N  e.  ZZ  ->  N  e.  CC )
2 mulcom 9595 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
31, 2sylan2 474 . . 3  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
43fveq2d 5876 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( exp `  ( N  x.  A
) ) )
5 oveq2 6304 . . . . . 6  |-  ( j  =  0  ->  ( A  x.  j )  =  ( A  x.  0 ) )
65fveq2d 5876 . . . . 5  |-  ( j  =  0  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  0 ) ) )
7 oveq2 6304 . . . . 5  |-  ( j  =  0  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ 0 ) )
86, 7eqeq12d 2479 . . . 4  |-  ( j  =  0  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) ) )
9 oveq2 6304 . . . . . 6  |-  ( j  =  k  ->  ( A  x.  j )  =  ( A  x.  k ) )
109fveq2d 5876 . . . . 5  |-  ( j  =  k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  k )
) )
11 oveq2 6304 . . . . 5  |-  ( j  =  k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ k
) )
1210, 11eqeq12d 2479 . . . 4  |-  ( j  =  k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  k
) )  =  ( ( exp `  A
) ^ k ) ) )
13 oveq2 6304 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A  x.  j )  =  ( A  x.  ( k  +  1 ) ) )
1413fveq2d 5876 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  ( k  +  1 ) ) ) )
15 oveq2 6304 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ (
k  +  1 ) ) )
1614, 15eqeq12d 2479 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  (
k  +  1 ) ) )  =  ( ( exp `  A
) ^ ( k  +  1 ) ) ) )
17 oveq2 6304 . . . . . 6  |-  ( j  =  -u k  ->  ( A  x.  j )  =  ( A  x.  -u k ) )
1817fveq2d 5876 . . . . 5  |-  ( j  =  -u k  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  -u k ) ) )
19 oveq2 6304 . . . . 5  |-  ( j  =  -u k  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ -u k
) )
2018, 19eqeq12d 2479 . . . 4  |-  ( j  =  -u k  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  -u k
) )  =  ( ( exp `  A
) ^ -u k
) ) )
21 oveq2 6304 . . . . . 6  |-  ( j  =  N  ->  ( A  x.  j )  =  ( A  x.  N ) )
2221fveq2d 5876 . . . . 5  |-  ( j  =  N  ->  ( exp `  ( A  x.  j ) )  =  ( exp `  ( A  x.  N )
) )
23 oveq2 6304 . . . . 5  |-  ( j  =  N  ->  (
( exp `  A
) ^ j )  =  ( ( exp `  A ) ^ N
) )
2422, 23eqeq12d 2479 . . . 4  |-  ( j  =  N  ->  (
( exp `  ( A  x.  j )
)  =  ( ( exp `  A ) ^ j )  <->  ( exp `  ( A  x.  N
) )  =  ( ( exp `  A
) ^ N ) ) )
25 ef0 13838 . . . . 5  |-  ( exp `  0 )  =  1
26 mul01 9776 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
2726fveq2d 5876 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( exp `  0
) )
28 efcl 13830 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2928exp0d 12307 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  A
) ^ 0 )  =  1 )
3025, 27, 293eqtr4a 2524 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( A  x.  0 ) )  =  ( ( exp `  A
) ^ 0 ) )
31 oveq1 6303 . . . . . . 7  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) )  =  ( ( ( exp `  A
) ^ k )  x.  ( exp `  A
) ) )
3231adantl 466 . . . . . 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 10826 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  k  e.  CC )
34 ax-1cn 9567 . . . . . . . . . . . 12  |-  1  e.  CC
35 adddi 9598 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC  /\  1  e.  CC )  ->  ( A  x.  ( k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
3634, 35mp3an3 1313 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  ( A  x.  1 ) ) )
37 mulid1 9610 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
3837adantr 465 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  1 )  =  A )
3938oveq2d 6312 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( ( A  x.  k )  +  ( A  x.  1 ) )  =  ( ( A  x.  k )  +  A ) )
4036, 39eqtrd 2498 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4133, 40sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  (
k  +  1 ) )  =  ( ( A  x.  k )  +  A ) )
4241fveq2d 5876 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( exp `  ( ( A  x.  k )  +  A
) ) )
43 mulcl 9593 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  k
)  e.  CC )
4433, 43sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A  x.  k
)  e.  CC )
45 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
46 efadd 13841 . . . . . . . . 9  |-  ( ( ( A  x.  k
)  e.  CC  /\  A  e.  CC )  ->  ( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4744, 45, 46syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  (
( A  x.  k
)  +  A ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4842, 47eqtrd 2498 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  ( A  x.  k ) )  x.  ( exp `  A
) ) )
4948adantr 465 . . . . . 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 12176 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5128, 50sylan 471 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ ( k  +  1 ) )  =  ( ( ( exp `  A ) ^ k )  x.  ( exp `  A
) ) )
5251adantr 465 . . . . . 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 2508 . . . . 5  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k ) )  ->  ( exp `  ( A  x.  ( k  +  1 ) ) )  =  ( ( exp `  A ) ^ ( k  +  1 ) ) )
5453exp31 604 . . . 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 6304 . . . . . 6  |-  ( ( exp `  ( A  x.  k ) )  =  ( ( exp `  A ) ^ k
)  ->  ( 1  /  ( exp `  ( A  x.  k )
) )  =  ( 1  /  ( ( exp `  A ) ^ k ) ) )
56 nncn 10564 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  CC )
57 mulneg2 10015 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  CC )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5856, 57sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  -u k
)  =  -u ( A  x.  k )
)
5958fveq2d 5876 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( exp `  -u ( A  x.  k ) ) )
6056, 43sylan2 474 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  x.  k
)  e.  CC )
61 efneg 13845 . . . . . . . . 9  |-  ( ( A  x.  k )  e.  CC  ->  ( exp `  -u ( A  x.  k ) )  =  ( 1  /  ( exp `  ( A  x.  k ) ) ) )
6260, 61syl 16 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  -u ( A  x.  k )
)  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
6359, 62eqtrd 2498 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( exp `  ( A  x.  -u k ) )  =  ( 1  /  ( exp `  ( A  x.  k )
) ) )
64 nnnn0 10823 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  NN0 )
65 expneg 12177 . . . . . . . 8  |-  ( ( ( exp `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6628, 64, 65syl2an 477 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  A
) ^ -u k
)  =  ( 1  /  ( ( exp `  A ) ^ k
) ) )
6763, 66eqeq12d 2479 . . . . . 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 221 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( exp `  ( A  x.  k )
)  =  ( ( exp `  A ) ^ k )  -> 
( exp `  ( A  x.  -u k ) )  =  ( ( exp `  A ) ^ -u k ) ) )
6968ex 434 . . . 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 10986 . . 3  |-  ( A  e.  CC  ->  ( N  e.  ZZ  ->  ( exp `  ( A  x.  N ) )  =  ( ( exp `  A ) ^ N
) ) )
7170imp 429 . 2  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( A  x.  N )
)  =  ( ( exp `  A ) ^ N ) )
724, 71eqtr3d 2500 1  |-  ( ( A  e.  CC  /\  N  e.  ZZ )  ->  ( exp `  ( N  x.  A )
)  =  ( ( exp `  A ) ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   -ucneg 9825    / cdiv 10227   NNcn 10556   NN0cn0 10816   ZZcz 10885   ^cexp 12169   expce 13809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815
This theorem is referenced by:  efzval  13849  efgt0  13850  tanval3  13881  demoivre  13947  ef2kpi  22997  efif1olem4  23058  explog  23104  reexplog  23105  relogexp  23106  tanarg  23130  root1eq1  23255
  Copyright terms: Public domain W3C validator