Theorem efexp 15983

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 12504	. . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
2		mulcom 11137	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
3	1, 2	sylan2 593	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴 · 𝑁) = (𝑁 · 𝐴))
4	3	fveq2d 6846	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = (exp‘(𝑁 · 𝐴)))
5		oveq2 7365	. . . . . 6 ⊢ (𝑗 = 0 → (𝐴 · 𝑗) = (𝐴 · 0))
6	5	fveq2d 6846	. . . . 5 ⊢ (𝑗 = 0 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 0)))
7		oveq2 7365	. . . . 5 ⊢ (𝑗 = 0 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑0))
8	6, 7	eqeq12d 2752	. . . 4 ⊢ (𝑗 = 0 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0)))
9		oveq2 7365	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐴 · 𝑗) = (𝐴 · 𝑘))
10	9	fveq2d 6846	. . . . 5 ⊢ (𝑗 = 𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑘)))
11		oveq2 7365	. . . . 5 ⊢ (𝑗 = 𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑘))
12	10, 11	eqeq12d 2752	. . . 4 ⊢ (𝑗 = 𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)))
13		oveq2 7365	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → (𝐴 · 𝑗) = (𝐴 · (𝑘 + 1)))
14	13	fveq2d 6846	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · (𝑘 + 1))))
15		oveq2 7365	. . . . 5 ⊢ (𝑗 = (𝑘 + 1) → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑(𝑘 + 1)))
16	14, 15	eqeq12d 2752	. . . 4 ⊢ (𝑗 = (𝑘 + 1) → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1))))
17		oveq2 7365	. . . . . 6 ⊢ (𝑗 = -𝑘 → (𝐴 · 𝑗) = (𝐴 · -𝑘))
18	17	fveq2d 6846	. . . . 5 ⊢ (𝑗 = -𝑘 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · -𝑘)))
19		oveq2 7365	. . . . 5 ⊢ (𝑗 = -𝑘 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑-𝑘))
20	18, 19	eqeq12d 2752	. . . 4 ⊢ (𝑗 = -𝑘 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · -𝑘)) = ((exp‘𝐴)↑-𝑘)))
21		oveq2 7365	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝐴 · 𝑗) = (𝐴 · 𝑁))
22	21	fveq2d 6846	. . . . 5 ⊢ (𝑗 = 𝑁 → (exp‘(𝐴 · 𝑗)) = (exp‘(𝐴 · 𝑁)))
23		oveq2 7365	. . . . 5 ⊢ (𝑗 = 𝑁 → ((exp‘𝐴)↑𝑗) = ((exp‘𝐴)↑𝑁))
24	22, 23	eqeq12d 2752	. . . 4 ⊢ (𝑗 = 𝑁 → ((exp‘(𝐴 · 𝑗)) = ((exp‘𝐴)↑𝑗) ↔ (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
25		ef0 15973	. . . . 5 ⊢ (exp‘0) = 1
26		mul01 11334	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
27	26	fveq2d 6846	. . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = (exp‘0))
28		efcl 15965	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
29	28	exp0d 14045	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((exp‘𝐴)↑0) = 1)
30	25, 27, 29	3eqtr4a 2802	. . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(𝐴 · 0)) = ((exp‘𝐴)↑0))
31		oveq1 7364	. . . . . . 7 ⊢ ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
32	31	adantl 482	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
33		nn0cn 12423	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
34		ax-1cn 11109	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
35		adddi 11140	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
36	34, 35	mp3an3 1450	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + (𝐴 · 1)))
37		mulid1 11153	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
38	37	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 1) = 𝐴)
39	38	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((𝐴 · 𝑘) + (𝐴 · 1)) = ((𝐴 · 𝑘) + 𝐴))
40	36, 39	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
41	33, 40	sylan2 593	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · (𝑘 + 1)) = ((𝐴 · 𝑘) + 𝐴))
42	41	fveq2d 6846	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = (exp‘((𝐴 · 𝑘) + 𝐴)))
43		mulcl 11135	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · 𝑘) ∈ ℂ)
44	33, 43	sylan2 593	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴 · 𝑘) ∈ ℂ)
45		simpl 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℂ)
46		efadd 15976	. . . . . . . . 9 ⊢ (((𝐴 · 𝑘) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
47	44, 45, 46	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘((𝐴 · 𝑘) + 𝐴)) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
48	42, 47	eqtrd 2776	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
49	48	adantr 481	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘(𝐴 · 𝑘)) · (exp‘𝐴)))
50		expp1 13974	. . . . . . . 8 ⊢ (((exp‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
51	28, 50	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑(𝑘 + 1)) = (((exp‘𝐴)↑𝑘) · (exp‘𝐴)))
52	51	adantr 481	. . . . . 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 2786	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) ∧ (exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘)) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))
54	53	exp31 420	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑘 ∈ ℕ0 → ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (exp‘(𝐴 · (𝑘 + 1))) = ((exp‘𝐴)↑(𝑘 + 1)))))
55		oveq2 7365	. . . . . 6 ⊢ ((exp‘(𝐴 · 𝑘)) = ((exp‘𝐴)↑𝑘) → (1 / (exp‘(𝐴 · 𝑘))) = (1 / ((exp‘𝐴)↑𝑘)))
56		nncn 12161	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
57		mulneg2 11592	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
58	56, 57	sylan2 593	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · -𝑘) = -(𝐴 · 𝑘))
59	58	fveq2d 6846	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (exp‘-(𝐴 · 𝑘)))
60	56, 43	sylan2 593	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐴 · 𝑘) ∈ ℂ)
61		efneg 15980	. . . . . . . . 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 2776	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (exp‘(𝐴 · -𝑘)) = (1 / (exp‘(𝐴 · 𝑘))))
64		nnnn0 12420	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
65		expneg 13975	. . . . . . . 8 ⊢ (((exp‘𝐴) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
66	28, 64, 65	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ) → ((exp‘𝐴)↑-𝑘) = (1 / ((exp‘𝐴)↑𝑘)))
67	63, 66	eqeq12d 2752	. . . . . 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 413	. . . 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 12604	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑁 ∈ ℤ → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁)))
71	70	imp 407	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝐴 · 𝑁)) = ((exp‘𝐴)↑𝑁))
72	4, 71	eqtr3d 2778	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (exp‘(𝑁 · 𝐴)) = ((exp‘𝐴)↑𝑁))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  -cneg 11386   / cdiv 11812  ℕcn 12153  ℕ₀cn0 12413  ℤcz 12499  ↑cexp 13967  expce 15944

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950

This theorem is referenced by:  efzval  15984  efgt0  15985  tanval3  16016  demoivre  16082  ef2kpi  25835  efif1olem4  25901  explog  25949  reexplog  25950  relogexp  25951  tanarg  25974  root1eq1  26108  vtsprod  33252