Theorem efexp 11623

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.

Step	Hyp	Ref	Expression
1		zcn 9196	. . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
2		mulcom 7882	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
3	1, 2	sylan2 284	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
4	3	fveq2d 5490	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = (exp‘(𝑁 · 𝐴)))
5		oveq2 5850	. . . . . 6 ⊢ (𝑗 = 0 → (𝐴 · 𝑗) = (𝐴 · 0))
6	5	fveq2d 5490	. . . . 5 ⊢ (𝑗 = 0 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 0)))
7		oveq2 5850	. . . . 5 ⊢ (𝑗 = 0 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑0))
8	6, 7	eqeq12d 2180	. . . 4 ⊢ (𝑗 = 0 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0)))
9		oveq2 5850	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴 · 𝑗) = (𝐴 · 𝑘))
10	9	fveq2d 5490	. . . . 5 ⊢ (𝑗 = 𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑘)))
11		oveq2 5850	. . . . 5 ⊢ (𝑗 = 𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑘))
12	10, 11	eqeq12d 2180	. . . 4 ⊢ (𝑗 = 𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)))
13		oveq2 5850	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (𝐴 · 𝑗) = (𝐴 · (𝑘 + 1)))
14	13	fveq2d 5490	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · (𝑘 + 1))))
15		oveq2 5850	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑(𝑘 + 1)))
16	14, 15	eqeq12d 2180	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1))))
17		oveq2 5850	. . . . . 6 ⊢ (𝑗 = -𝑘 → (𝐴 · 𝑗) = (𝐴 · -𝑘))
18	17	fveq2d 5490	. . . . 5 ⊢ (𝑗 = -𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · -𝑘)))
19		oveq2 5850	. . . . 5 ⊢ (𝑗 = -𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑-𝑘))
20	18, 19	eqeq12d 2180	. . . 4 ⊢ (𝑗 = -𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
21		oveq2 5850	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴 · 𝑗) = (𝐴 · 𝑁))
22	21	fveq2d 5490	. . . . 5 ⊢ (𝑗 = 𝑁 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑁)))
23		oveq2 5850	. . . . 5 ⊢ (𝑗 = 𝑁 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑁))
24	22, 23	eqeq12d 2180	. . . 4 ⊢ (𝑗 = 𝑁 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
25		ef0 11613	. . . . 5 ⊢ (exp‘0) = 1
26		mul01 8287	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
27	26	fveq2d 5490	. . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = (exp‘0))
28		efcl 11605	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
29	28	exp0d 10582	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴)↑0) = 1)
30	25, 27, 29	3eqtr4a 2225	. . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0))
31		oveq1 5849	. . . . . . 7 ⊢ ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
32	31	adantl 275	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
33		nn0cn 9124	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
34		ax-1cn 7846	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
35		adddi 7885	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
36	34, 35	mp3an3 1316	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
37		mulid1 7896	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
38	37	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 1) = 𝐴)
39	38	oveq2d 5858	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝐴 · 𝑘) + (𝐴 · 1)) = ((𝐴 · 𝑘) + 𝐴))
40	36, 39	eqtrd 2198	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
41	33, 40	sylan2 284	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
42	41	fveq2d 5490	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = (exp‘((𝐴 · 𝑘) + 𝐴)))
43		mulcl 7880	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 𝑘) ∈ ℂ)
44	33, 43	sylan2 284	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · 𝑘) ∈ ℂ)
45		simpl 108	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)
46		efadd 11616	. . . . . . . . 9 ⊢ (((𝐴 · 𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
47	44, 45, 46	syl2anc 409	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
48	42, 47	eqtrd 2198	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
49	48	adantr 274	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
50		expp1 10462	. . . . . . . 8 ⊢ (((exp‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
51	28, 50	sylan 281	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
52	51	adantr 274	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
53	32, 49, 52	3eqtr4d 2208	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))
54	53	exp31 362	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℕ0 → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))))
55		oveq2 5850	. . . . . 6 ⊢ ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘)))
56		nncn 8865	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
57		mulneg2 8294	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
58	56, 57	sylan2 284	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
59	58	fveq2d 5490	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (exp‘-(𝐴 · 𝑘)))
60	56, 43	sylan2 284	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · 𝑘) ∈ ℂ)
61		efneg 11620	. . . . . . . . 9 ⊢ ((𝐴 · 𝑘) ∈ ℂ → (exp‘-(𝐴 · 𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
62	60, 61	syl 14	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘-(𝐴 · 𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
63	59, 62	eqtrd 2198	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
64		efap0 11618	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) # 0)
65		nnnn0 9121	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
66		expnegap0 10463	. . . . . . . 8 ⊢ (((exp‘𝐴) ∈ ℂ ∧ (exp‘𝐴) # 0 ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
67	28, 64, 65, 66	syl2an3an 1288	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
68	63, 67	eqeq12d 2180	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘) ↔ (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘))))
69	55, 68	syl5ibr 155	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
70	69	ex 114	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℕ → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘))))
71	8, 12, 16, 20, 24, 30, 54, 70	zindd 9309	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑁 ∈ ℤ → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
72	71	imp 123	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁))
73	4, 72	eqtr3d 2200	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · 𝐴)) = ((exp‘𝐴)↑𝑁))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136   class class class wbr 3982  ‘cfv 5188  (class class class)co 5842  ℂcc 7751  0cc0 7753  1c1 7754   + caddc 7756   · cmul 7758  -cneg 8070   # cap 8479   / cdiv 8568  ℕcn 8857  ℕ₀cn0 9114  ℤcz 9191  ↑cexp 10454  expce 11583

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589

This theorem is referenced by:  efzval  11624  efgt0  11625  tanval3ap  11655  demoivre  11713  ef2kpi  13367  reexplog  13432  relogexp  13433