Theorem efexp 15738

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 12254	. . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
2		mulcom 10888	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
3	1, 2	sylan2 592	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
4	3	fveq2d 6760	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = (exp‘(𝑁 · 𝐴)))
5		oveq2 7263	. . . . . 6 ⊢ (𝑗 = 0 → (𝐴 · 𝑗) = (𝐴 · 0))
6	5	fveq2d 6760	. . . . 5 ⊢ (𝑗 = 0 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 0)))
7		oveq2 7263	. . . . 5 ⊢ (𝑗 = 0 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑0))
8	6, 7	eqeq12d 2754	. . . 4 ⊢ (𝑗 = 0 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0)))
9		oveq2 7263	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴 · 𝑗) = (𝐴 · 𝑘))
10	9	fveq2d 6760	. . . . 5 ⊢ (𝑗 = 𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑘)))
11		oveq2 7263	. . . . 5 ⊢ (𝑗 = 𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑘))
12	10, 11	eqeq12d 2754	. . . 4 ⊢ (𝑗 = 𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)))
13		oveq2 7263	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (𝐴 · 𝑗) = (𝐴 · (𝑘 + 1)))
14	13	fveq2d 6760	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · (𝑘 + 1))))
15		oveq2 7263	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑(𝑘 + 1)))
16	14, 15	eqeq12d 2754	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1))))
17		oveq2 7263	. . . . . 6 ⊢ (𝑗 = -𝑘 → (𝐴 · 𝑗) = (𝐴 · -𝑘))
18	17	fveq2d 6760	. . . . 5 ⊢ (𝑗 = -𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · -𝑘)))
19		oveq2 7263	. . . . 5 ⊢ (𝑗 = -𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑-𝑘))
20	18, 19	eqeq12d 2754	. . . 4 ⊢ (𝑗 = -𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
21		oveq2 7263	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴 · 𝑗) = (𝐴 · 𝑁))
22	21	fveq2d 6760	. . . . 5 ⊢ (𝑗 = 𝑁 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑁)))
23		oveq2 7263	. . . . 5 ⊢ (𝑗 = 𝑁 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑁))
24	22, 23	eqeq12d 2754	. . . 4 ⊢ (𝑗 = 𝑁 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
25		ef0 15728	. . . . 5 ⊢ (exp‘0) = 1
26		mul01 11084	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
27	26	fveq2d 6760	. . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = (exp‘0))
28		efcl 15720	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
29	28	exp0d 13786	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴)↑0) = 1)
30	25, 27, 29	3eqtr4a 2805	. . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0))
31		oveq1 7262	. . . . . . 7 ⊢ ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
32	31	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
33		nn0cn 12173	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
34		ax-1cn 10860	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
35		adddi 10891	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
36	34, 35	mp3an3 1448	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
37		mulid1 10904	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
38	37	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 1) = 𝐴)
39	38	oveq2d 7271	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝐴 · 𝑘) + (𝐴 · 1)) = ((𝐴 · 𝑘) + 𝐴))
40	36, 39	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
41	33, 40	sylan2 592	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
42	41	fveq2d 6760	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = (exp‘((𝐴 · 𝑘) + 𝐴)))
43		mulcl 10886	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 𝑘) ∈ ℂ)
44	33, 43	sylan2 592	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · 𝑘) ∈ ℂ)
45		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)
46		efadd 15731	. . . . . . . . 9 ⊢ (((𝐴 · 𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
47	44, 45, 46	syl2anc 583	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
48	42, 47	eqtrd 2778	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
49	48	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
50		expp1 13717	. . . . . . . 8 ⊢ (((exp‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
51	28, 50	sylan 579	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
52	51	adantr 480	. . . . . 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 2788	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))
54	53	exp31 419	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℕ0 → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))))
55		oveq2 7263	. . . . . 6 ⊢ ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘)))
56		nncn 11911	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
57		mulneg2 11342	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
58	56, 57	sylan2 592	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
59	58	fveq2d 6760	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (exp‘-(𝐴 · 𝑘)))
60	56, 43	sylan2 592	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · 𝑘) ∈ ℂ)
61		efneg 15735	. . . . . . . . 9 ⊢ ((𝐴 · 𝑘) ∈ ℂ → (exp‘-(𝐴 · 𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
62	60, 61	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘-(𝐴 · 𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
63	59, 62	eqtrd 2778	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
64		nnnn0 12170	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
65		expneg 13718	. . . . . . . 8 ⊢ (((exp‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
66	28, 64, 65	syl2an 595	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
67	63, 66	eqeq12d 2754	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘) ↔ (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘))))
68	55, 67	syl5ibr 245	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
69	68	ex 412	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℕ → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘))))
70	8, 12, 16, 20, 24, 30, 54, 69	zindd 12351	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑁 ∈ ℤ → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
71	70	imp 406	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁))
72	4, 71	eqtr3d 2780	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · 𝐴)) = ((exp‘𝐴)↑𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807  -cneg 11136   / cdiv 11562  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ↑cexp 13710  expce 15699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-ico 13014  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-fac 13916  df-bc 13945  df-hash 13973  df-shft 14706  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-limsup 15108  df-clim 15125  df-rlim 15126  df-sum 15326  df-ef 15705

This theorem is referenced by:  efzval  15739  efgt0  15740  tanval3  15771  demoivre  15837  ef2kpi  25540  efif1olem4  25606  explog  25654  reexplog  25655  relogexp  25656  tanarg  25679  root1eq1  25813  vtsprod  32519