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Theorem efexp 11674
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 9247 . . . 4 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
2 mulcom 7931 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
31, 2sylan2 286 . . 3 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
43fveq2d 5515 . 2 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = (exp‘(𝑁 · 𝐴)))
5 oveq2 5877 . . . . . 6 (𝑗 = 0 → (𝐴 · 𝑗) = (𝐴 · 0))
65fveq2d 5515 . . . . 5 (𝑗 = 0 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 0)))
7 oveq2 5877 . . . . 5 (𝑗 = 0 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑0))
86, 7eqeq12d 2192 . . . 4 (𝑗 = 0 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0)))
9 oveq2 5877 . . . . . 6 (𝑗 = 𝑘 → (𝐴 · 𝑗) = (𝐴 · 𝑘))
109fveq2d 5515 . . . . 5 (𝑗 = 𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑘)))
11 oveq2 5877 . . . . 5 (𝑗 = 𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑘))
1210, 11eqeq12d 2192 . . . 4 (𝑗 = 𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)))
13 oveq2 5877 . . . . . 6 (𝑗 = (𝑘 + 1) → (𝐴 · 𝑗) = (𝐴 · (𝑘 + 1)))
1413fveq2d 5515 . . . . 5 (𝑗 = (𝑘 + 1) → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · (𝑘 + 1))))
15 oveq2 5877 . . . . 5 (𝑗 = (𝑘 + 1) → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑(𝑘 + 1)))
1614, 15eqeq12d 2192 . . . 4 (𝑗 = (𝑘 + 1) → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1))))
17 oveq2 5877 . . . . . 6 (𝑗 = -𝑘 → (𝐴 · 𝑗) = (𝐴 · -𝑘))
1817fveq2d 5515 . . . . 5 (𝑗 = -𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · -𝑘)))
19 oveq2 5877 . . . . 5 (𝑗 = -𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑-𝑘))
2018, 19eqeq12d 2192 . . . 4 (𝑗 = -𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
21 oveq2 5877 . . . . . 6 (𝑗 = 𝑁 → (𝐴 · 𝑗) = (𝐴 · 𝑁))
2221fveq2d 5515 . . . . 5 (𝑗 = 𝑁 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑁)))
23 oveq2 5877 . . . . 5 (𝑗 = 𝑁 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑁))
2422, 23eqeq12d 2192 . . . 4 (𝑗 = 𝑁 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
25 ef0 11664 . . . . 5 (exp‘0) = 1
26 mul01 8336 . . . . . 6 (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
2726fveq2d 5515 . . . . 5 (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = (exp‘0))
28 efcl 11656 . . . . . 6 (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
2928exp0d 10633 . . . . 5 (𝐴 ∈ ℂ → ((exp‘𝐴)↑0) = 1)
3025, 27, 293eqtr4a 2236 . . . 4 (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0))
31 oveq1 5876 . . . . . . 7 ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
3231adantl 277 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
33 nn0cn 9175 . . . . . . . . . 10 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
34 ax-1cn 7895 . . . . . . . . . . . 12 1 ∈ ℂ
35 adddi 7934 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
3634, 35mp3an3 1326 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
37 mulid1 7945 . . . . . . . . . . . . 13 (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
3837adantr 276 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 1) = 𝐴)
3938oveq2d 5885 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝐴 · 𝑘) + (𝐴 · 1)) = ((𝐴 · 𝑘) + 𝐴))
4036, 39eqtrd 2210 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
4133, 40sylan2 286 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
4241fveq2d 5515 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = (exp‘((𝐴 · 𝑘) + 𝐴)))
43 mulcl 7929 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 𝑘) ∈ ℂ)
4433, 43sylan2 286 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · 𝑘) ∈ ℂ)
45 simpl 109 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)
46 efadd 11667 . . . . . . . . 9 (((𝐴 · 𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
4744, 45, 46syl2anc 411 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
4842, 47eqtrd 2210 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
4948adantr 276 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
50 expp1 10513 . . . . . . . 8 (((exp‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
5128, 50sylan 283 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
5251adantr 276 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
5332, 49, 523eqtr4d 2220 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))
5453exp31 364 . . . 4 (𝐴 ∈ ℂ → (𝑘 ∈ ℕ0 → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))))
55 oveq2 5877 . . . . . 6 ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘)))
56 nncn 8916 . . . . . . . . . 10 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
57 mulneg2 8343 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
5856, 57sylan2 286 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
5958fveq2d 5515 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (exp‘-(𝐴 · 𝑘)))
6056, 43sylan2 286 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · 𝑘) ∈ ℂ)
61 efneg 11671 . . . . . . . . 9 ((𝐴 · 𝑘) ∈ ℂ → (exp‘-(𝐴 · 𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
6260, 61syl 14 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘-(𝐴 · 𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
6359, 62eqtrd 2210 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
64 efap0 11669 . . . . . . . 8 (𝐴 ∈ ℂ → (exp‘𝐴) # 0)
65 nnnn0 9172 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
66 expnegap0 10514 . . . . . . . 8 (((exp‘𝐴) ∈ ℂ ∧ (exp‘𝐴) # 0 ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
6728, 64, 65, 66syl2an3an 1298 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
6863, 67eqeq12d 2192 . . . . . 6 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘) ↔ (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘))))
6955, 68syl5ibr 156 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
7069ex 115 . . . 4 (𝐴 ∈ ℂ → (𝑘 ∈ ℕ → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘))))
718, 12, 16, 20, 24, 30, 54, 70zindd 9360 . . 3 (𝐴 ∈ ℂ → (𝑁 ∈ ℤ → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
7271imp 124 . 2 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁))
734, 72eqtr3d 2212 1 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · 𝐴)) = ((exp‘𝐴)↑𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148   class class class wbr 4000  cfv 5212  (class class class)co 5869  cc 7800  0cc0 7802  1c1 7803   + caddc 7805   · cmul 7807  -cneg 8119   # cap 8528   / cdiv 8618  cn 8908  0cn0 9165  cz 9242  cexp 10505  expce 11634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-ico 9881  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-bc 10712  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-ef 11640
This theorem is referenced by:  efzval  11675  efgt0  11676  tanval3ap  11706  demoivre  11764  ef2kpi  13894  reexplog  13959  relogexp  13960
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