efi4p - Intuitionistic Logic Explorer

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Theorem efi4p 12486

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 8274	. . . 4 ⊢ i ∈ ℂ
2		mulcl 8306	. . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
3	1, 2	mpan 428	. . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ)
4		efi4p.1	. . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ (((i · 𝐴)↑𝑛) / (!‘𝑛)))
5	4	ef4p 12463	. . 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 8272	. . . . . 6 ⊢ 1 ∈ ℂ
8		addcl 8304	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
9	7, 3, 8	sylancr 418	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 + (i · 𝐴)) ∈ ℂ)
10	3	sqcld 11111	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) ∈ ℂ)
11	10	halfcld 9552	. . . . 5 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) ∈ ℂ)
12		3nn0 9583	. . . . . . 7 ⊢ 3 ∈ ℕ₀
13		expcl 10996	. . . . . . 7 ⊢ (((i · 𝐴) ∈ ℂ ∧ 3 ∈ ℕ₀) → ((i · 𝐴)↑3) ∈ ℂ)
14	3, 12, 13	sylancl 417	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) ∈ ℂ)
15		6cn 9387	. . . . . . 7 ⊢ 6 ∈ ℂ
16		6re 9386	. . . . . . . 8 ⊢ 6 ∈ ℝ
17		6pos 9406	. . . . . . . 8 ⊢ 0 < 6
18	16, 17	gt0ap0ii 8957	. . . . . . 7 ⊢ 6 # 0
19		divclap 9009	. . . . . . 7 ⊢ ((((i · 𝐴)↑3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 # 0) → (((i · 𝐴)↑3) / 6) ∈ ℂ)
20	15, 18, 19	mp3an23 1370	. . . . . 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 8348	. . . 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 8495	. . . 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 9582	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
26		mulexp 11017	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 2 ∈ ℕ₀) → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
27	1, 25, 26	mp3an13 1369	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = ((i↑2) · (𝐴↑2)))
28		i2 11079	. . . . . . . . . . . 12 ⊢ (i↑2) = -1
29	28	oveq1i 6095	. . . . . . . . . . 11 ⊢ ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2))
30	29	a1i 9	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i↑2) · (𝐴↑2)) = (-1 · (𝐴↑2)))
31		sqcl 11039	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
32	31	mulm1d 8737	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-1 · (𝐴↑2)) = -(𝐴↑2))
33	27, 30, 32	3eqtrd 2275	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑2) = -(𝐴↑2))
34	33	oveq1d 6100	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) = (-(𝐴↑2) / 2))
35		2cn 9376	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
36		2ap0 9398	. . . . . . . . . 10 ⊢ 2 # 0
37		divnegap 9037	. . . . . . . . . 10 ⊢ (((𝐴↑2) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -((𝐴↑2) / 2) = (-(𝐴↑2) / 2))
38	35, 36, 37	mp3an23 1370	. . . . . . . . 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 2274	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑2) / 2) = -((𝐴↑2) / 2))
41	40	oveq2d 6101	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 + (((i · 𝐴)↑2) / 2)) = (1 + -((𝐴↑2) / 2)))
42	31	halfcld 9552	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ)
43		negsub 8574	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ ((𝐴↑2) / 2) ∈ ℂ) → (1 + -((𝐴↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
44	7, 42, 43	sylancr 418	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 + -((𝐴↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
45	41, 44	eqtrd 2271	. . . . 5 ⊢ (𝐴 ∈ ℂ → (1 + (((i · 𝐴)↑2) / 2)) = (1 − ((𝐴↑2) / 2)))
46		mulexp 11017	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 3 ∈ ℕ₀) → ((i · 𝐴)↑3) = ((i↑3) · (𝐴↑3)))
47	1, 12, 46	mp3an13 1369	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) = ((i↑3) · (𝐴↑3)))
48		i3 11080	. . . . . . . . . . 11 ⊢ (i↑3) = -i
49	48	oveq1i 6095	. . . . . . . . . 10 ⊢ ((i↑3) · (𝐴↑3)) = (-i · (𝐴↑3))
50	47, 49	eqtrdi 2287	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴)↑3) = (-i · (𝐴↑3)))
51	50	oveq1d 6100	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) = ((-i · (𝐴↑3)) / 6))
52		expcl 10996	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝐴↑3) ∈ ℂ)
53	12, 52	mpan2 429	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑3) ∈ ℂ)
54		negicn 8527	. . . . . . . . . 10 ⊢ -i ∈ ℂ
55	15, 18	pm3.2i 272	. . . . . . . . . 10 ⊢ (6 ∈ ℂ ∧ 6 # 0)
56		divassap 9021	. . . . . . . . . 10 ⊢ ((-i ∈ ℂ ∧ (𝐴↑3) ∈ ℂ ∧ (6 ∈ ℂ ∧ 6 # 0)) → ((-i · (𝐴↑3)) / 6) = (-i · ((𝐴↑3) / 6)))
57	54, 55, 56	mp3an13 1369	. . . . . . . . 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 9009	. . . . . . . . . . 11 ⊢ (((𝐴↑3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 6 # 0) → ((𝐴↑3) / 6) ∈ ℂ)
60	15, 18, 59	mp3an23 1370	. . . . . . . . . 10 ⊢ ((𝐴↑3) ∈ ℂ → ((𝐴↑3) / 6) ∈ ℂ)
61	53, 60	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((𝐴↑3) / 6) ∈ ℂ)
62		mulneg12 8724	. . . . . . . . 9 ⊢ ((i ∈ ℂ ∧ ((𝐴↑3) / 6) ∈ ℂ) → (-i · ((𝐴↑3) / 6)) = (i · -((𝐴↑3) / 6)))
63	1, 61, 62	sylancr 418	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (-i · ((𝐴↑3) / 6)) = (i · -((𝐴↑3) / 6)))
64	51, 58, 63	3eqtrd 2275	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((i · 𝐴)↑3) / 6) = (i · -((𝐴↑3) / 6)))
65	64	oveq2d 6101	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (((i · 𝐴)↑3) / 6)) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
66	61	negcld 8624	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → -((𝐴↑3) / 6) ∈ ℂ)
67		adddi 8311	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ -((𝐴↑3) / 6) ∈ ℂ) → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
68	1, 67	mp3an1 1365	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ -((𝐴↑3) / 6) ∈ ℂ) → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
69	66, 68	mpdan 425	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (𝐴 + -((𝐴↑3) / 6))) = ((i · 𝐴) + (i · -((𝐴↑3) / 6))))
70		negsub 8574	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ ((𝐴↑3) / 6) ∈ ℂ) → (𝐴 + -((𝐴↑3) / 6)) = (𝐴 − ((𝐴↑3) / 6)))
71	61, 70	mpdan 425	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 + -((𝐴↑3) / 6)) = (𝐴 − ((𝐴↑3) / 6)))
72	71	oveq2d 6101	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (𝐴 + -((𝐴↑3) / 6))) = (i · (𝐴 − ((𝐴↑3) / 6))))
73	65, 69, 72	3eqtr2d 2277	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) + (((i · 𝐴)↑3) / 6)) = (i · (𝐴 − ((𝐴↑3) / 6))))
74	45, 73	oveq12d 6103	. . . 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 2275	. . 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 6100	. 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 2271	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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 class class class wbr 4130 ↦ cmpt 4192 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 0cc0 8179 1c1 8180 ici 8181 + caddc 8182 · cmul 8184 − cmin 8497 -cneg 8498 # cap 8910 / cdiv 9003 2c2 9356 3c3 9357 4c4 9358 6c6 9360 ℕ₀cn0 9565 ℤ_≥cuz 9923 ↑cexp 10977 !cfa 11165 Σcsu 12121 expce 12411

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-5 9367 df-6 9368 df-n0 9566 df-z 9647 df-uz 9924 df-q 10022 df-rp 10057 df-ico 10298 df-fz 10414 df-fzo 10552 df-seqfrec 10887 df-exp 10978 df-fac 11166 df-ihash 11217 df-cj 11609 df-re 11610 df-im 11611 df-rsqrt 11766 df-abs 11767 df-clim 12047 df-sumdc 12122 df-ef 12417

This theorem is used by: resin4p 12487 recos4p 12488

W3C validator