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Theorem efexp 15446
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 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · 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 11974 . . . 4 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
2 mulcom 10612 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
31, 2sylan2 595 . . 3 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
43fveq2d 6649 . 2 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = (exp‘(𝑁 · 𝐴)))
5 oveq2 7143 . . . . . 6 (𝑗 = 0 → (𝐴 · 𝑗) = (𝐴 · 0))
65fveq2d 6649 . . . . 5 (𝑗 = 0 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 0)))
7 oveq2 7143 . . . . 5 (𝑗 = 0 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑0))
86, 7eqeq12d 2814 . . . 4 (𝑗 = 0 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0)))
9 oveq2 7143 . . . . . 6 (𝑗 = 𝑘 → (𝐴 · 𝑗) = (𝐴 · 𝑘))
109fveq2d 6649 . . . . 5 (𝑗 = 𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑘)))
11 oveq2 7143 . . . . 5 (𝑗 = 𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑘))
1210, 11eqeq12d 2814 . . . 4 (𝑗 = 𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)))
13 oveq2 7143 . . . . . 6 (𝑗 = (𝑘 + 1) → (𝐴 · 𝑗) = (𝐴 · (𝑘 + 1)))
1413fveq2d 6649 . . . . 5 (𝑗 = (𝑘 + 1) → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · (𝑘 + 1))))
15 oveq2 7143 . . . . 5 (𝑗 = (𝑘 + 1) → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑(𝑘 + 1)))
1614, 15eqeq12d 2814 . . . 4 (𝑗 = (𝑘 + 1) → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1))))
17 oveq2 7143 . . . . . 6 (𝑗 = -𝑘 → (𝐴 · 𝑗) = (𝐴 · -𝑘))
1817fveq2d 6649 . . . . 5 (𝑗 = -𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · -𝑘)))
19 oveq2 7143 . . . . 5 (𝑗 = -𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑-𝑘))
2018, 19eqeq12d 2814 . . . 4 (𝑗 = -𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
21 oveq2 7143 . . . . . 6 (𝑗 = 𝑁 → (𝐴 · 𝑗) = (𝐴 · 𝑁))
2221fveq2d 6649 . . . . 5 (𝑗 = 𝑁 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑁)))
23 oveq2 7143 . . . . 5 (𝑗 = 𝑁 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑁))
2422, 23eqeq12d 2814 . . . 4 (𝑗 = 𝑁 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
25 ef0 15436 . . . . 5 (exp‘0) = 1
26 mul01 10808 . . . . . 6 (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
2726fveq2d 6649 . . . . 5 (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = (exp‘0))
28 efcl 15428 . . . . . 6 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
2928exp0d 13500 . . . . 5 (𝐴 ∈ ℂ → ((exp‘𝐴)↑0) = 1)
3025, 27, 293eqtr4a 2859 . . . 4 (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0))
31 oveq1 7142 . . . . . . 7 ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
3231adantl 485 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
33 nn0cn 11895 . . . . . . . . . 10 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
34 ax-1cn 10584 . . . . . . . . . . . 12 1 ∈ ℂ
35 adddi 10615 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
3634, 35mp3an3 1447 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
37 mulid1 10628 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
3837adantr 484 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 1) = 𝐴)
3938oveq2d 7151 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝐴 · 𝑘) + (𝐴 · 1)) = ((𝐴 · 𝑘) + 𝐴))
4036, 39eqtrd 2833 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
4133, 40sylan2 595 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
4241fveq2d 6649 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = (exp‘((𝐴 · 𝑘) + 𝐴)))
43 mulcl 10610 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 𝑘) ∈ ℂ)
4433, 43sylan2 595 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · 𝑘) ∈ ℂ)
45 simpl 486 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)
46 efadd 15439 . . . . . . . . 9 (((𝐴 · 𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
4744, 45, 46syl2anc 587 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
4842, 47eqtrd 2833 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
4948adantr 484 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
50 expp1 13432 . . . . . . . 8 (((exp‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
5128, 50sylan 583 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
5251adantr 484 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
5332, 49, 523eqtr4d 2843 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))
5453exp31 423 . . . 4 (𝐴 ∈ ℂ → (𝑘 ∈ ℕ0 → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))))
55 oveq2 7143 . . . . . 6 ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘)))
56 nncn 11633 . . . . . . . . . 10 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
57 mulneg2 11066 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
5856, 57sylan2 595 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
5958fveq2d 6649 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (exp‘-(𝐴 · 𝑘)))
6056, 43sylan2 595 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · 𝑘) ∈ ℂ)
61 efneg 15443 . . . . . . . . 9 ((𝐴 · 𝑘) ∈ ℂ → (exp‘-(𝐴 · 𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
6260, 61syl 17 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘-(𝐴 · 𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
6359, 62eqtrd 2833 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
64 nnnn0 11892 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
65 expneg 13433 . . . . . . . 8 (((exp‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
6628, 64, 65syl2an 598 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
6763, 66eqeq12d 2814 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘) ↔ (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘))))
6855, 67syl5ibr 249 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
6968ex 416 . . . 4 (𝐴 ∈ ℂ → (𝑘 ∈ ℕ → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘))))
708, 12, 16, 20, 24, 30, 54, 69zindd 12071 . . 3 (𝐴 ∈ ℂ → (𝑁 ∈ ℤ → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
7170imp 410 . 2 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁))
724, 71eqtr3d 2835 1 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · 𝐴)) = ((exp‘𝐴)↑𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  cfv 6324  (class class class)co 7135  cc 10524  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  -cneg 10860   / cdiv 11286  cn 11625  0cn0 11885  cz 11969  cexp 13425  expce 15407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-inf 8891  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-shft 14418  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-ef 15413
This theorem is referenced by:  efzval  15447  efgt0  15448  tanval3  15479  demoivre  15545  ef2kpi  25071  efif1olem4  25137  explog  25185  reexplog  25186  relogexp  25187  tanarg  25210  root1eq1  25344  vtsprod  32020
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