efi4p - Intuitionistic Logic Explorer

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Theorem efi4p 12271

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 8120	. . . 4 ⊢ i ∈ ℂ
2		mulcl 8152	. . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
3	1, 2	mpan 424	. . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ)
4		efi4p.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))
5	4	ef4p 12248	. . 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 14	. 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 8118	. . . . . 6 ⊢ 1 ∈ ℂ
8		addcl 8150	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
9	7, 3, 8	sylancr 414	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 + (i · 𝐴)) ∈ ℂ)
10	3	sqcld 10926	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) ∈ ℂ)
11	10	halfcld 9382	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) ∈ ℂ)
12		3nn0 9413	. . . . . . 7 ⊢ 3 ∈ ℕ₀
13		expcl 10812	. . . . . . 7 ⊢ (((i · 𝐴) ∈ ℂ ∧ 3 ∈ ℕ₀) → ((i · 𝐴)↑3) ∈ ℂ)
14	3, 12, 13	sylancl 413	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) ∈ ℂ)
15		6cn 9218	. . . . . . 7 ⊢ 6 ∈ ℂ
16		6re 9217	. . . . . . . 8 ⊢ 6 ∈ ℝ
17		6pos 9237	. . . . . . . 8 ⊢ 0 < 6
18	16, 17	gt0ap0ii 8801	. . . . . . 7 ⊢ 6 # 0
19		divclap 8851	. . . . . . 7 ⊢ ((((i · 𝐴)↑3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 # 0) → (((i · 𝐴)↑3) / 6) ∈ ℂ)
20	15, 18, 19	mp3an23 1363	. . . . . 6 ⊢ (((i · 𝐴)↑3) ∈ ℂ → (((i · 𝐴)↑3) / 6) ∈ ℂ)
21	14, 20	syl 14	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) ∈ ℂ)
22	9, 11, 21	addassd 8195	. . . 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 9	. . . . 5 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ)
24	23, 3, 11, 21	add4d 8341	. . . 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 9412	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
26		mulexp 10833	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 2 ∈ ℕ₀) → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
27	1, 25, 26	mp3an13 1362	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
28		i2 10895	. . . . . . . . . . . 12 ⊢ (i↑2) = -1
29	28	oveq1i 6023	. . . . . . . . . . 11 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2))
30	29	a1i 9	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)))
31		sqcl 10855	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
32	31	mulm1d 8582	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2))
33	27, 30, 32	3eqtrd 2266	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2))
34	33	oveq1d 6028	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) = (-(𝐴↑2) / 2))
35		2cn 9207	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
36		2ap0 9229	. . . . . . . . . 10 ⊢ 2 # 0
37		divnegap 8879	. . . . . . . . . 10 ⊢ (((𝐴↑2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -((𝐴↑2) / 2) = (-(𝐴↑2) / 2))
38	35, 36, 37	mp3an23 1363	. . . . . . . . 9 ⊢ ((𝐴↑2) ∈ ℂ → -((𝐴↑2) / 2) = (-(𝐴↑2) / 2))
39	31, 38	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → -((𝐴↑2) / 2) = (-(𝐴↑2) / 2))
40	34, 39	eqtr4d 2265	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) = -((𝐴↑2) / 2))
41	40	oveq2d 6029	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 + (((i · 𝐴)↑2) / 2)) = (1 + -((𝐴↑2) / 2)))
42	31	halfcld 9382	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ)
43		negsub 8420	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ ((𝐴↑2) / 2) ∈ ℂ) → (1 + -((𝐴↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
44	7, 42, 43	sylancr 414	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 + -((𝐴↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
45	41, 44	eqtrd 2262	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 + (((i · 𝐴)↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
46		mulexp 10833	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 3 ∈ ℕ₀) → ((i · 𝐴)↑3) = ((i↑3) · (𝐴↑3)))
47	1, 12, 46	mp3an13 1362	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) = ((i↑3) · (𝐴↑3)))
48		i3 10896	. . . . . . . . . . 11 ⊢ (i↑3) = -i
49	48	oveq1i 6023	. . . . . . . . . 10 ⊢ ((i↑3) · (𝐴↑3)) = (-i · (𝐴↑3))
50	47, 49	eqtrdi 2278	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) = (-i · (𝐴↑3)))
51	50	oveq1d 6028	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) = ((-i · (𝐴↑3)) / 6))
52		expcl 10812	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐴↑3) ∈ ℂ)
53	12, 52	mpan2 425	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑3) ∈ ℂ)
54		negicn 8373	. . . . . . . . . 10 ⊢ -i ∈ ℂ
55	15, 18	pm3.2i 272	. . . . . . . . . 10 ⊢ (6 ∈ ℂ ∧ 6 # 0)
56		divassap 8863	. . . . . . . . . 10 ⊢ ((-i ∈ ℂ ∧ (𝐴↑3) ∈ ℂ ∧ (6 ∈ ℂ ∧ 6 # 0)) → ((-i · (𝐴↑3)) / 6) = (-i · ((𝐴↑3) / 6)))
57	54, 55, 56	mp3an13 1362	. . . . . . . . 9 ⊢ ((𝐴↑3) ∈ ℂ → ((-i · (𝐴↑3)) / 6) = (-i · ((𝐴↑3) / 6)))
58	53, 57	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((-i · (𝐴↑3)) / 6) = (-i · ((𝐴↑3) / 6)))
59		divclap 8851	. . . . . . . . . . 11 ⊢ (((𝐴↑3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 # 0) → ((𝐴↑3) / 6) ∈ ℂ)
60	15, 18, 59	mp3an23 1363	. . . . . . . . . 10 ⊢ ((𝐴↑3) ∈ ℂ → ((𝐴↑3) / 6) ∈ ℂ)
61	53, 60	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝐴↑3) / 6) ∈ ℂ)
62		mulneg12 8569	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ ((𝐴↑3) / 6) ∈ ℂ) → (-i · ((𝐴↑3) / 6)) = (i · -((𝐴↑3) / 6)))
63	1, 61, 62	sylancr 414	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · ((𝐴↑3) / 6)) = (i · -((𝐴↑3) / 6)))
64	51, 58, 63	3eqtrd 2266	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) = (i · -((𝐴↑3) / 6)))
65	64	oveq2d 6029	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (((i · 𝐴)↑3) / 6)) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
66	61	negcld 8470	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → -((𝐴↑3) / 6) ∈ ℂ)
67		adddi 8157	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ -((𝐴↑3) / 6) ∈ ℂ) → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
68	1, 67	mp3an1 1358	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ -((𝐴↑3) / 6) ∈ ℂ) → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
69	66, 68	mpdan 421	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
70		negsub 8420	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ ((𝐴↑3) / 6) ∈ ℂ) → (𝐴 + -((𝐴↑3) / 6)) = (𝐴 − ((𝐴↑3) / 6)))
71	61, 70	mpdan 421	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 + -((𝐴↑3) / 6)) = (𝐴 − ((𝐴↑3) / 6)))
72	71	oveq2d 6029	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (𝐴 + -((𝐴↑3) / 6))) = (i · (𝐴 − ((𝐴↑3) / 6))))
73	65, 69, 72	3eqtr2d 2268	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (((i · 𝐴)↑3) / 6)) = (i · (𝐴 − ((𝐴↑3) / 6))))
74	45, 73	oveq12d 6031	. . . 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 2266	. . 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 6028	. 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 2262	1 ⊢ (𝐴 ∈ ℂ → (exp‘(i · 𝐴)) = (((1 − ((𝐴↑2) / 2)) + (i · (𝐴 − ((𝐴↑3) / 6)))) + Σ𝑘 ∈ (ℤ_≥‘4)(𝐹‘𝑘)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 class class class wbr 4086 ↦ cmpt 4148 ‘cfv 5324 (class class class)co 6013 ℂcc 8023 0cc0 8025 1c1 8026 ici 8027 + caddc 8028 · cmul 8030 − cmin 8343 -cneg 8344 # cap 8754 / cdiv 8845 2c2 9187 3c3 9188 4c4 9189 6c6 9191 ℕ₀cn0 9395 ℤ_≥cuz 9748 ↑cexp 10793 !cfa 10980 Σcsu 11907 expce 12196

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-6 9199 df-n0 9396 df-z 9473 df-uz 9749 df-q 9847 df-rp 9882 df-ico 10122 df-fz 10237 df-fzo 10371 df-seqfrec 10703 df-exp 10794 df-fac 10981 df-ihash 11031 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-sumdc 11908 df-ef 12202

This theorem is referenced by: resin4p 12272 recos4p 12273

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