efi4p - Intuitionistic Logic Explorer

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Theorem efi4p 11062

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 7494	. . . 4 ⊢ i ∈ ℂ
2		mulcl 7523	. . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
3	1, 2	mpan 416	. . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ)
4		efi4p.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))
5	4	ef4p 11038	. . 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 7492	. . . . . 6 ⊢ 1 ∈ ℂ
8		addcl 7521	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
9	7, 3, 8	sylancr 406	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 + (i · 𝐴)) ∈ ℂ)
10	3	sqcld 10138	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) ∈ ℂ)
11	10	halfcld 8714	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) ∈ ℂ)
12		3nn0 8745	. . . . . . 7 ⊢ 3 ∈ ℕ₀
13		expcl 10027	. . . . . . 7 ⊢ (((i · 𝐴) ∈ ℂ ∧ 3 ∈ ℕ₀) → ((i · 𝐴)↑3) ∈ ℂ)
14	3, 12, 13	sylancl 405	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) ∈ ℂ)
15		6cn 8558	. . . . . . 7 ⊢ 6 ∈ ℂ
16		6re 8557	. . . . . . . 8 ⊢ 6 ∈ ℝ
17		6pos 8577	. . . . . . . 8 ⊢ 0 < 6
18	16, 17	gt0ap0ii 8158	. . . . . . 7 ⊢ 6 # 0
19		divclap 8199	. . . . . . 7 ⊢ ((((i · 𝐴)↑3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 # 0) → (((i · 𝐴)↑3) / 6) ∈ ℂ)
20	15, 18, 19	mp3an23 1266	. . . . . 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 7564	. . . 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 7705	. . . 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 8744	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
26		mulexp 10048	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 2 ∈ ℕ₀) → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
27	1, 25, 26	mp3an13 1265	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
28		i2 10109	. . . . . . . . . . . 12 ⊢ (i↑2) = -1
29	28	oveq1i 5676	. . . . . . . . . . 11 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2))
30	29	a1i 9	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)))
31		sqcl 10070	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
32	31	mulm1d 7942	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2))
33	27, 30, 32	3eqtrd 2125	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2))
34	33	oveq1d 5681	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) = (-(𝐴↑2) / 2))
35		2cn 8547	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
36		2ap0 8569	. . . . . . . . . 10 ⊢ 2 # 0
37		divnegap 8227	. . . . . . . . . 10 ⊢ (((𝐴↑2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -((𝐴↑2) / 2) = (-(𝐴↑2) / 2))
38	35, 36, 37	mp3an23 1266	. . . . . . . . 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 2124	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) = -((𝐴↑2) / 2))
41	40	oveq2d 5682	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 + (((i · 𝐴)↑2) / 2)) = (1 + -((𝐴↑2) / 2)))
42	31	halfcld 8714	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ)
43		negsub 7784	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ ((𝐴↑2) / 2) ∈ ℂ) → (1 + -((𝐴↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
44	7, 42, 43	sylancr 406	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 + -((𝐴↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
45	41, 44	eqtrd 2121	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 + (((i · 𝐴)↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
46		mulexp 10048	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 3 ∈ ℕ₀) → ((i · 𝐴)↑3) = ((i↑3) · (𝐴↑3)))
47	1, 12, 46	mp3an13 1265	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) = ((i↑3) · (𝐴↑3)))
48		i3 10110	. . . . . . . . . . 11 ⊢ (i↑3) = -i
49	48	oveq1i 5676	. . . . . . . . . 10 ⊢ ((i↑3) · (𝐴↑3)) = (-i · (𝐴↑3))
50	47, 49	syl6eq 2137	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) = (-i · (𝐴↑3)))
51	50	oveq1d 5681	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) = ((-i · (𝐴↑3)) / 6))
52		expcl 10027	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐴↑3) ∈ ℂ)
53	12, 52	mpan2 417	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑3) ∈ ℂ)
54		negicn 7737	. . . . . . . . . 10 ⊢ -i ∈ ℂ
55	15, 18	pm3.2i 267	. . . . . . . . . 10 ⊢ (6 ∈ ℂ ∧ 6 # 0)
56		divassap 8211	. . . . . . . . . 10 ⊢ ((-i ∈ ℂ ∧ (𝐴↑3) ∈ ℂ ∧ (6 ∈ ℂ ∧ 6 # 0)) → ((-i · (𝐴↑3)) / 6) = (-i · ((𝐴↑3) / 6)))
57	54, 55, 56	mp3an13 1265	. . . . . . . . 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 8199	. . . . . . . . . . 11 ⊢ (((𝐴↑3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 # 0) → ((𝐴↑3) / 6) ∈ ℂ)
60	15, 18, 59	mp3an23 1266	. . . . . . . . . 10 ⊢ ((𝐴↑3) ∈ ℂ → ((𝐴↑3) / 6) ∈ ℂ)
61	53, 60	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝐴↑3) / 6) ∈ ℂ)
62		mulneg12 7929	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ ((𝐴↑3) / 6) ∈ ℂ) → (-i · ((𝐴↑3) / 6)) = (i · -((𝐴↑3) / 6)))
63	1, 61, 62	sylancr 406	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · ((𝐴↑3) / 6)) = (i · -((𝐴↑3) / 6)))
64	51, 58, 63	3eqtrd 2125	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) = (i · -((𝐴↑3) / 6)))
65	64	oveq2d 5682	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (((i · 𝐴)↑3) / 6)) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
66	61	negcld 7834	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → -((𝐴↑3) / 6) ∈ ℂ)
67		adddi 7528	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ -((𝐴↑3) / 6) ∈ ℂ) → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
68	1, 67	mp3an1 1261	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ -((𝐴↑3) / 6) ∈ ℂ) → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
69	66, 68	mpdan 413	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
70		negsub 7784	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ ((𝐴↑3) / 6) ∈ ℂ) → (𝐴 + -((𝐴↑3) / 6)) = (𝐴 − ((𝐴↑3) / 6)))
71	61, 70	mpdan 413	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 + -((𝐴↑3) / 6)) = (𝐴 − ((𝐴↑3) / 6)))
72	71	oveq2d 5682	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (𝐴 + -((𝐴↑3) / 6))) = (i · (𝐴 − ((𝐴↑3) / 6))))
73	65, 69, 72	3eqtr2d 2127	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (((i · 𝐴)↑3) / 6)) = (i · (𝐴 − ((𝐴↑3) / 6))))
74	45, 73	oveq12d 5684	. . . 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 2125	. . 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 5681	. 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 2121	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 103 = wceq 1290 ∈ wcel 1439 class class class wbr 3851 ↦ cmpt 3905 ‘cfv 5028 (class class class)co 5666 ℂcc 7402 0cc0 7404 1c1 7405 ici 7406 + caddc 7407 · cmul 7409 − cmin 7707 -cneg 7708 # cap 8112 / cdiv 8193 2c2 8527 3c3 8528 4c4 8529 6c6 8531 ℕ₀cn0 8727 ℤ_≥cuz 9073 ↑cexp 10008 !cfa 10187 Σcsu 10796 expce 10986

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 ax-arch 7518 ax-caucvg 7519

This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-isom 5037 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-frec 6170 df-1o 6195 df-oadd 6199 df-er 6306 df-en 6512 df-dom 6513 df-fin 6514 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-3 8536 df-4 8537 df-5 8538 df-6 8539 df-n0 8728 df-z 8805 df-uz 9074 df-q 9159 df-rp 9189 df-ico 9366 df-fz 9479 df-fzo 9608 df-iseq 9907 df-seq3 9908 df-exp 10009 df-fac 10188 df-ihash 10238 df-cj 10330 df-re 10331 df-im 10332 df-rsqrt 10485 df-abs 10486 df-clim 10721 df-isum 10797 df-ef 10992

This theorem is referenced by: resin4p 11063 recos4p 11064

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