efi4p - Metamath Proof Explorer

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Theorem efi4p 16112

Description: Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.)

**Hypothesis**
Ref	Expression
efi4p.1	⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))

**Assertion**
Ref	Expression
efi4p	⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ_≥‘4)(𝐹‘𝑘)))

Distinct variable groups: 𝐴,𝑘,𝑛 𝑘,𝐹 Allowed substitution hint: 𝐹(𝑛)

**Proof of Theorem efi4p**
Step	Hyp	Ref	Expression
1		ax-icn 11134	. . . 4 ⊢ i ∈ ℂ
2		mulcl 11159	. . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
3	1, 2	mpan 690	. . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ)
4		efi4p.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))
5	4	ef4p 16088	. . 3 ⊢ ((i · 𝐴) ∈ ℂ → (exp‘(i · 𝐴)) = ((((1 + (i · 𝐴)) + (((i · 𝐴)↑2) / 2)) + (((i · 𝐴)↑3) / 6)) + Σ𝑘 ∈ (ℤ_≥‘4)(𝐹‘𝑘)))
6	3, 5	syl 17	. 2 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = ((((1 + (i · 𝐴)) + (((i · 𝐴)↑2) / 2)) + (((i · 𝐴)↑3) / 6)) + Σ𝑘 ∈ (ℤ_≥‘4)(𝐹‘𝑘)))
7		ax-1cn 11133	. . . . . 6 ⊢ 1 ∈ ℂ
8		addcl 11157	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
9	7, 3, 8	sylancr 587	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 + (i · 𝐴)) ∈ ℂ)
10	3	sqcld 14116	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) ∈ ℂ)
11	10	halfcld 12434	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) ∈ ℂ)
12		3nn0 12467	. . . . . . 7 ⊢ 3 ∈ ℕ₀
13		expcl 14051	. . . . . . 7 ⊢ (((i · 𝐴) ∈ ℂ ∧ 3 ∈ ℕ₀) → ((i · 𝐴)↑3) ∈ ℂ)
14	3, 12, 13	sylancl 586	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) ∈ ℂ)
15		6cn 12284	. . . . . . 7 ⊢ 6 ∈ ℂ
16		6re 12283	. . . . . . . 8 ⊢ 6 ∈ ℝ
17		6pos 12303	. . . . . . . 8 ⊢ 0 < 6
18	16, 17	gt0ne0ii 11721	. . . . . . 7 ⊢ 6 ≠ 0
19		divcl 11850	. . . . . . 7 ⊢ ((((i · 𝐴)↑3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 ≠ 0) → (((i · 𝐴)↑3) / 6) ∈ ℂ)
20	15, 18, 19	mp3an23 1455	. . . . . 6 ⊢ (((i · 𝐴)↑3) ∈ ℂ → (((i · 𝐴)↑3) / 6) ∈ ℂ)
21	14, 20	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) ∈ ℂ)
22	9, 11, 21	addassd 11203	. . . 4 ⊢ (𝐴 ∈ ℂ → (((1 + (i · 𝐴)) + (((i · 𝐴)↑2) / 2)) + (((i · 𝐴)↑3) / 6)) = ((1 + (i · 𝐴)) + ((((i · 𝐴)↑2) / 2) + (((i · 𝐴)↑3) / 6))))
23	7	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ)
24	23, 3, 11, 21	add4d 11410	. . . 4 ⊢ (𝐴 ∈ ℂ → ((1 + (i · 𝐴)) + ((((i · 𝐴)↑2) / 2) + (((i · 𝐴)↑3) / 6))) = ((1 + (((i · 𝐴)↑2) / 2)) + ((i · 𝐴) + (((i · 𝐴)↑3) / 6))))
25		2nn0 12466	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
26		mulexp 14073	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 2 ∈ ℕ₀) → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
27	1, 25, 26	mp3an13 1454	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
28		i2 14174	. . . . . . . . . . . 12 ⊢ (i↑2) = -1
29	28	oveq1i 7400	. . . . . . . . . . 11 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2))
30	29	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)))
31		sqcl 14090	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
32	31	mulm1d 11637	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2))
33	27, 30, 32	3eqtrd 2769	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2))
34	33	oveq1d 7405	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) = (-(𝐴↑2) / 2))
35		2cn 12268	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
36		2ne0 12297	. . . . . . . . . 10 ⊢ 2 ≠ 0
37		divneg 11881	. . . . . . . . . 10 ⊢ (((𝐴↑2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → -((𝐴↑2) / 2) = (-(𝐴↑2) / 2))
38	35, 36, 37	mp3an23 1455	. . . . . . . . 9 ⊢ ((𝐴↑2) ∈ ℂ → -((𝐴↑2) / 2) = (-(𝐴↑2) / 2))
39	31, 38	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → -((𝐴↑2) / 2) = (-(𝐴↑2) / 2))
40	34, 39	eqtr4d 2768	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) = -((𝐴↑2) / 2))
41	40	oveq2d 7406	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 + (((i · 𝐴)↑2) / 2)) = (1 + -((𝐴↑2) / 2)))
42	31	halfcld 12434	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ)
43		negsub 11477	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ ((𝐴↑2) / 2) ∈ ℂ) → (1 + -((𝐴↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
44	7, 42, 43	sylancr 587	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 + -((𝐴↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
45	41, 44	eqtrd 2765	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 + (((i · 𝐴)↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
46		mulexp 14073	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 3 ∈ ℕ₀) → ((i · 𝐴)↑3) = ((i↑3) · (𝐴↑3)))
47	1, 12, 46	mp3an13 1454	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) = ((i↑3) · (𝐴↑3)))
48		i3 14175	. . . . . . . . . . 11 ⊢ (i↑3) = -i
49	48	oveq1i 7400	. . . . . . . . . 10 ⊢ ((i↑3) · (𝐴↑3)) = (-i · (𝐴↑3))
50	47, 49	eqtrdi 2781	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) = (-i · (𝐴↑3)))
51	50	oveq1d 7405	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) = ((-i · (𝐴↑3)) / 6))
52		expcl 14051	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐴↑3) ∈ ℂ)
53	12, 52	mpan2 691	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑3) ∈ ℂ)
54		negicn 11429	. . . . . . . . . 10 ⊢ -i ∈ ℂ
55	15, 18	pm3.2i 470	. . . . . . . . . 10 ⊢ (6 ∈ ℂ ∧ 6 ≠ 0)
56		divass 11862	. . . . . . . . . 10 ⊢ ((-i ∈ ℂ ∧ (𝐴↑3) ∈ ℂ ∧ (6 ∈ ℂ ∧ 6 ≠ 0)) → ((-i · (𝐴↑3)) / 6) = (-i · ((𝐴↑3) / 6)))
57	54, 55, 56	mp3an13 1454	. . . . . . . . 9 ⊢ ((𝐴↑3) ∈ ℂ → ((-i · (𝐴↑3)) / 6) = (-i · ((𝐴↑3) / 6)))
58	53, 57	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((-i · (𝐴↑3)) / 6) = (-i · ((𝐴↑3) / 6)))
59		divcl 11850	. . . . . . . . . . 11 ⊢ (((𝐴↑3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 ≠ 0) → ((𝐴↑3) / 6) ∈ ℂ)
60	15, 18, 59	mp3an23 1455	. . . . . . . . . 10 ⊢ ((𝐴↑3) ∈ ℂ → ((𝐴↑3) / 6) ∈ ℂ)
61	53, 60	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝐴↑3) / 6) ∈ ℂ)
62		mulneg12 11623	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ ((𝐴↑3) / 6) ∈ ℂ) → (-i · ((𝐴↑3) / 6)) = (i · -((𝐴↑3) / 6)))
63	1, 61, 62	sylancr 587	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · ((𝐴↑3) / 6)) = (i · -((𝐴↑3) / 6)))
64	51, 58, 63	3eqtrd 2769	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) = (i · -((𝐴↑3) / 6)))
65	64	oveq2d 7406	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (((i · 𝐴)↑3) / 6)) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
66	61	negcld 11527	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → -((𝐴↑3) / 6) ∈ ℂ)
67		adddi 11164	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ -((𝐴↑3) / 6) ∈ ℂ) → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
68	1, 67	mp3an1 1450	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ -((𝐴↑3) / 6) ∈ ℂ) → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
69	66, 68	mpdan 687	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
70		negsub 11477	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ ((𝐴↑3) / 6) ∈ ℂ) → (𝐴 + -((𝐴↑3) / 6)) = (𝐴 − ((𝐴↑3) / 6)))
71	61, 70	mpdan 687	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 + -((𝐴↑3) / 6)) = (𝐴 − ((𝐴↑3) / 6)))
72	71	oveq2d 7406	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (𝐴 + -((𝐴↑3) / 6))) = (i · (𝐴 − ((𝐴↑3) / 6))))
73	65, 69, 72	3eqtr2d 2771	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (((i · 𝐴)↑3) / 6)) = (i · (𝐴 − ((𝐴↑3) / 6))))
74	45, 73	oveq12d 7408	. . . 4 ⊢ (𝐴 ∈ ℂ → ((1 + (((i · 𝐴)↑2) / 2)) + ((i · 𝐴) + (((i · 𝐴)↑3) / 6))) = ((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))))
75	22, 24, 74	3eqtrd 2769	. . 3 ⊢ (𝐴 ∈ ℂ → (((1 + (i · 𝐴)) + (((i · 𝐴)↑2) / 2)) + (((i · 𝐴)↑3) / 6)) = ((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))))
76	75	oveq1d 7405	. 2 ⊢ (𝐴 ∈ ℂ → ((((1 + (i · 𝐴)) + (((i · 𝐴)↑2) / 2)) + (((i · 𝐴)↑3) / 6)) + Σ𝑘 ∈ (ℤ_≥‘4)(𝐹‘𝑘)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ_≥‘4)(𝐹‘𝑘)))
77	6, 76	eqtrd 2765	1 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ_≥‘4)(𝐹‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 0cc0 11075 1c1 11076 ici 11077 + caddc 11078 · cmul 11080 − cmin 11412 -cneg 11413 / cdiv 11842 2c2 12248 3c3 12249 4c4 12250 6c6 12252 ℕ₀cn0 12449 ℤ_≥cuz 12800 ↑cexp 14033 !cfa 14245 Σcsu 15659 expce 16034

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-ico 13319 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-fac 14246 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040

This theorem is referenced by: resin4p 16113 recos4p 16114

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