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Theorem dveflem 15750

Description: Derivative of the exponential function at 0. The key step in the proof is eftlub 12435, to show that abs(exp(𝑥) − 1 − 𝑥) ≤ abs(𝑥)↑2 · (3 / 4). (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

**Assertion**
Ref	Expression
dveflem	⊢ 0(ℂ D exp)1

Proof of Theorem dveflem Dummy variables 𝑘 𝑛 𝑤 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 8308	. . 3 ⊢ 0 ∈ ℂ
2		eqid 2238	. . . . 5 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
3	2	cntoptop 15557	. . . 4 ⊢ (MetOpen‘(abs ∘ − )) ∈ Top
4		unicntopcntop 15566	. . . . 5 ⊢ ℂ = ∪ (MetOpen‘(abs ∘ − ))
5	4	ntrtop 15152	. . . 4 ⊢ ((MetOpen‘(abs ∘ − )) ∈ Top → ((int‘(MetOpen‘(abs ∘ − )))‘ℂ) = ℂ)
6	3, 5	ax-mp 5	. . 3 ⊢ ((int‘(MetOpen‘(abs ∘ − )))‘ℂ) = ℂ
7	1, 6	eleqtrri 2314	. 2 ⊢ 0 ∈ ((int‘(MetOpen‘(abs ∘ − )))‘ℂ)
8		ax-1cn 8262	. . 3 ⊢ 1 ∈ ℂ
9		1rp 10037	. . . . . 6 ⊢ 1 ∈ ℝ⁺
10		rpmincl 11982	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → inf({𝑥, 1}, ℝ, < ) ∈ ℝ⁺)
11	9, 10	mpan2 429	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → inf({𝑥, 1}, ℝ, < ) ∈ ℝ⁺)
12		breq1 4128	. . . . . . . 8 ⊢ (𝑢 = 𝑤 → (𝑢 # 0 ↔ 𝑤 # 0))
13	12	elrab 2982	. . . . . . 7 ⊢ (𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↔ (𝑤 ∈ ℂ ∧ 𝑤 # 0))
14		simprl 535	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → 𝑤 ∈ ℂ)
15	14	subid1d 8616	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → (𝑤 − 0) = 𝑤)
16	15	fveq2d 5694	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → (abs‘(𝑤 − 0)) = (abs‘𝑤))
17	16	breq1d 4135	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → ((abs‘(𝑤 − 0)) < inf({𝑥, 1}, ℝ, < ) ↔ (abs‘𝑤) < inf({𝑥, 1}, ℝ, < )))
18	14	abscld 11925	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → (abs‘𝑤) ∈ ℝ)
19		rpre 10040	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
20	19	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → 𝑥 ∈ ℝ)
21		1red 8331	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → 1 ∈ ℝ)
22		ltmininf 11979	. . . . . . . . . . 11 ⊢ (((abs‘𝑤) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘𝑤) < inf({𝑥, 1}, ℝ, < ) ↔ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)))
23	18, 20, 21, 22	syl3anc 1278	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → ((abs‘𝑤) < inf({𝑥, 1}, ℝ, < ) ↔ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)))
24	17, 23	bitrd 188	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → ((abs‘(𝑤 − 0)) < inf({𝑥, 1}, ℝ, < ) ↔ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)))
25		eqid 2238	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)) = (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))
26		fveq2 5690	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (exp‘𝑧) = (exp‘𝑤))
27	26	oveq1d 6090	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((exp‘𝑧) − 1) = ((exp‘𝑤) − 1))
28		id 19	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
29	27, 28	oveq12d 6093	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (((exp‘𝑧) − 1) / 𝑧) = (((exp‘𝑤) − 1) / 𝑤))
30		simplr 533	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (𝑤 ∈ ℂ ∧ 𝑤 # 0))
31	30, 13	sylibr 134	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → 𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0})
32		efcl 12409	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (exp‘𝑤) ∈ ℂ)
33		peano2cnm 8582	. . . . . . . . . . . . . . . 16 ⊢ ((exp‘𝑤) ∈ ℂ → ((exp‘𝑤) − 1) ∈ ℂ)
34	14, 32, 33	3syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → ((exp‘𝑤) − 1) ∈ ℂ)
35		simprr 537	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → 𝑤 # 0)
36	34, 14, 35	divclapd 9110	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → (((exp‘𝑤) − 1) / 𝑤) ∈ ℂ)
37	36	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (((exp‘𝑤) − 1) / 𝑤) ∈ ℂ)
38	25, 29, 31, 37	fvmptd3 5793	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → ((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) = (((exp‘𝑤) − 1) / 𝑤))
39	38	fvoveq1d 6097	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) = (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1)))
40		1cnd 8332	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → 1 ∈ ℂ)
41	37, 40	subcld 8627	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → ((((exp‘𝑤) − 1) / 𝑤) − 1) ∈ ℂ)
42	41	abscld 11925	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1)) ∈ ℝ)
43		simplrl 541	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → 𝑤 ∈ ℂ)
44	43	abscld 11925	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (abs‘𝑤) ∈ ℝ)
45		simpll 531	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → 𝑥 ∈ ℝ⁺)
46	45	rpred 10076	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → 𝑥 ∈ ℝ)
47		abscl 11795	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℂ → (abs‘𝑤) ∈ ℝ)
48	47	ad2antrr 492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘𝑤) ∈ ℝ)
49	32	ad2antrr 492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (exp‘𝑤) ∈ ℂ)
50		subcl 8515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((exp‘𝑤) ∈ ℂ ∧ 1 ∈ ℂ) → ((exp‘𝑤) − 1) ∈ ℂ)
51	49, 8, 50	sylancl 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((exp‘𝑤) − 1) ∈ ℂ)
52		simpll 531	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → 𝑤 ∈ ℂ)
53		simplr 533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → 𝑤 # 0)
54	51, 52, 53	divclapd 9110	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (((exp‘𝑤) − 1) / 𝑤) ∈ ℂ)
55		1cnd 8332	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → 1 ∈ ℂ)
56	54, 55	subcld 8627	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((((exp‘𝑤) − 1) / 𝑤) − 1) ∈ ℂ)
57	56	abscld 11925	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1)) ∈ ℝ)
58	48, 57	remulcld 8346	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘𝑤) · (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1))) ∈ ℝ)
59	48	resqcld 11115	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘𝑤)↑2) ∈ ℝ)
60		3re 9357	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
61		4nn 9447	. . . . . . . . . . . . . . . . . 18 ⊢ 4 ∈ ℕ
62		nndivre 9319	. . . . . . . . . . . . . . . . . 18 ⊢ ((3 ∈ ℝ ∧ 4 ∈ ℕ) → (3 / 4) ∈ ℝ)
63	60, 61, 62	mp2an 430	. . . . . . . . . . . . . . . . 17 ⊢ (3 / 4) ∈ ℝ
64		remulcl 8297	. . . . . . . . . . . . . . . . 17 ⊢ ((((abs‘𝑤)↑2) ∈ ℝ ∧ (3 / 4) ∈ ℝ) → (((abs‘𝑤)↑2) · (3 / 4)) ∈ ℝ)
65	59, 63, 64	sylancl 417	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (((abs‘𝑤)↑2) · (3 / 4)) ∈ ℝ)
66	51, 52	subcld 8627	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (((exp‘𝑤) − 1) − 𝑤) ∈ ℂ)
67	66, 52, 53	divcanap2d 9112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (𝑤 · ((((exp‘𝑤) − 1) − 𝑤) / 𝑤)) = (((exp‘𝑤) − 1) − 𝑤))
68	51, 52, 52, 53	divsubdirapd 9150	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((((exp‘𝑤) − 1) − 𝑤) / 𝑤) = ((((exp‘𝑤) − 1) / 𝑤) − (𝑤 / 𝑤)))
69	52, 53	dividapd 9106	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (𝑤 / 𝑤) = 1)
70	69	oveq2d 6091	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((((exp‘𝑤) − 1) / 𝑤) − (𝑤 / 𝑤)) = ((((exp‘𝑤) − 1) / 𝑤) − 1))
71	68, 70	eqtrd 2271	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((((exp‘𝑤) − 1) − 𝑤) / 𝑤) = ((((exp‘𝑤) − 1) / 𝑤) − 1))
72	71	oveq2d 6091	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (𝑤 · ((((exp‘𝑤) − 1) − 𝑤) / 𝑤)) = (𝑤 · ((((exp‘𝑤) − 1) / 𝑤) − 1)))
73	49, 55, 52	subsub4d 8658	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (((exp‘𝑤) − 1) − 𝑤) = ((exp‘𝑤) − (1 + 𝑤)))
74		addcl 8294	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (1 + 𝑤) ∈ ℂ)
75	8, 52, 74	sylancr 418	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (1 + 𝑤) ∈ ℂ)
76		2nn0 9559	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 ∈ ℕ₀
77		eqid 2238	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))
78	77	eftlcl 12433	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑤 ∈ ℂ ∧ 2 ∈ ℕ₀) → Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘) ∈ ℂ)
79	52, 76, 78	sylancl 417	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘) ∈ ℂ)
80		df-2 9342	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 2 = (1 + 1)
81		1nn0 9558	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℕ₀
82		1e0p1 9797	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 = (0 + 1)
83		0nn0 9557	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℕ₀
84		0cnd 8309	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → 0 ∈ ℂ)
85	77	efval2 12410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 ∈ ℂ → (exp‘𝑤) = Σ𝑘 ∈ ℕ₀ ((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘))
86	85	ad2antrr 492	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (exp‘𝑤) = Σ𝑘 ∈ ℕ₀ ((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘))
87		nn0uz 9936	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ℕ₀ = (ℤ_≥‘0)
88	87	sumeq1i 12107	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Σ𝑘 ∈ ℕ₀ ((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘0)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘)
89	86, 88	eqtr2di 2288	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → Σ𝑘 ∈ (ℤ_≥‘0)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘) = (exp‘𝑤))
90	89	oveq2d 6091	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (0 + Σ𝑘 ∈ (ℤ_≥‘0)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘)) = (0 + (exp‘𝑤)))
91	49	addlidd 8466	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (0 + (exp‘𝑤)) = (exp‘𝑤))
92	90, 91	eqtr2d 2272	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (exp‘𝑤) = (0 + Σ𝑘 ∈ (ℤ_≥‘0)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘)))
93		eft0val 12438	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑤 ∈ ℂ → ((𝑤↑0) / (!‘0)) = 1)
94	93	ad2antrr 492	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((𝑤↑0) / (!‘0)) = 1)
95	94	oveq2d 6091	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (0 + ((𝑤↑0) / (!‘0))) = (0 + 1))
96	95, 82	eqtr4di 2289	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (0 + ((𝑤↑0) / (!‘0))) = 1)
97	77, 82, 83, 52, 84, 92, 96	efsep 12436	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (exp‘𝑤) = (1 + Σ𝑘 ∈ (ℤ_≥‘1)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘)))
98		exp1 10960	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 ∈ ℂ → (𝑤↑1) = 𝑤)
99	98	ad2antrr 492	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (𝑤↑1) = 𝑤)
100	99	oveq1d 6090	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((𝑤↑1) / (!‘1)) = (𝑤 / (!‘1)))
101		fac1 11145	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (!‘1) = 1
102	101	oveq2i 6086	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 / (!‘1)) = (𝑤 / 1)
103	100, 102	eqtrdi 2287	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((𝑤↑1) / (!‘1)) = (𝑤 / 1))
104		div1 9023	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 ∈ ℂ → (𝑤 / 1) = 𝑤)
105	104	ad2antrr 492	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (𝑤 / 1) = 𝑤)
106	103, 105	eqtrd 2271	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((𝑤↑1) / (!‘1)) = 𝑤)
107	106	oveq2d 6091	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (1 + ((𝑤↑1) / (!‘1))) = (1 + 𝑤))
108	77, 80, 81, 52, 55, 97, 107	efsep 12436	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (exp‘𝑤) = ((1 + 𝑤) + Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘)))
109	75, 79, 108	mvrladdd 8683	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((exp‘𝑤) − (1 + 𝑤)) = Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘))
110	73, 109	eqtrd 2271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (((exp‘𝑤) − 1) − 𝑤) = Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘))
111	67, 72, 110	3eqtr3d 2279	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (𝑤 · ((((exp‘𝑤) − 1) / 𝑤) − 1)) = Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘))
112	111	fveq2d 5694	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘(𝑤 · ((((exp‘𝑤) − 1) / 𝑤) − 1))) = (abs‘Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘)))
113	52, 56	absmuld 11938	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘(𝑤 · ((((exp‘𝑤) − 1) / 𝑤) − 1))) = ((abs‘𝑤) · (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1))))
114	112, 113	eqtr3d 2273	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘)) = ((abs‘𝑤) · (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1))))
115		eqid 2238	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ ↦ (((abs‘𝑤)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ (((abs‘𝑤)↑𝑛) / (!‘𝑛)))
116		eqid 2238	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ₀ ↦ ((((abs‘𝑤)↑2) / (!‘2)) · ((1 / (2 + 1))↑𝑛))) = (𝑛 ∈ ℕ₀ ↦ ((((abs‘𝑤)↑2) / (!‘2)) · ((1 / (2 + 1))↑𝑛)))
117		2nn 9445	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
118	117	a1i 9	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → 2 ∈ ℕ)
119		1red 8331	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → 1 ∈ ℝ)
120		simpr 110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘𝑤) < 1)
121	48, 119, 120	ltled 8435	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘𝑤) ≤ 1)
122	77, 115, 116, 118, 52, 121	eftlub 12435	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘Σ𝑘 ∈ (ℤ_≥‘2)((𝑛 ∈ ℕ₀ ↦ ((𝑤↑𝑛) / (!‘𝑛)))‘𝑘)) ≤ (((abs‘𝑤)↑2) · ((2 + 1) / ((!‘2) · 2))))
123	114, 122	eqbrtrrd 4149	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘𝑤) · (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1))) ≤ (((abs‘𝑤)↑2) · ((2 + 1) / ((!‘2) · 2))))
124		df-3 9343	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 = (2 + 1)
125		fac2 11147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (!‘2) = 2
126	125	oveq1i 6085	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((!‘2) · 2) = (2 · 2)
127		2t2e4 9438	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · 2) = 4
128	126, 127	eqtr2i 2260	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 = ((!‘2) · 2)
129	124, 128	oveq12i 6087	. . . . . . . . . . . . . . . . . 18 ⊢ (3 / 4) = ((2 + 1) / ((!‘2) · 2))
130	129	oveq2i 6086	. . . . . . . . . . . . . . . . 17 ⊢ (((abs‘𝑤)↑2) · (3 / 4)) = (((abs‘𝑤)↑2) · ((2 + 1) / ((!‘2) · 2)))
131	123, 130	breqtrrdi 4167	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘𝑤) · (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1))) ≤ (((abs‘𝑤)↑2) · (3 / 4)))
132	63	a1i 9	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (3 / 4) ∈ ℝ)
133	48	sqge0d 11116	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → 0 ≤ ((abs‘𝑤)↑2))
134		1re 8315	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ
135		3lt4 9456	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 < 4
136		4cn 9361	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 4 ∈ ℂ
137	136	mulridi 8318	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (4 · 1) = 4
138	135, 137	breqtrri 4152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 3 < (4 · 1)
139		4re 9360	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 4 ∈ ℝ
140		4pos 9380	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 < 4
141	139, 140	pm3.2i 272	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (4 ∈ ℝ ∧ 0 < 4)
142		ltdivmul 9196	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((3 ∈ ℝ ∧ 1 ∈ ℝ ∧ (4 ∈ ℝ ∧ 0 < 4)) → ((3 / 4) < 1 ↔ 3 < (4 · 1)))
143	60, 134, 141, 142	mp3an 1378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((3 / 4) < 1 ↔ 3 < (4 · 1))
144	138, 143	mpbir 146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 / 4) < 1
145	63, 134, 144	ltleii 8418	. . . . . . . . . . . . . . . . . . 19 ⊢ (3 / 4) ≤ 1
146	145	a1i 9	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (3 / 4) ≤ 1)
147	132, 119, 59, 133, 146	lemul2ad 9260	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (((abs‘𝑤)↑2) · (3 / 4)) ≤ (((abs‘𝑤)↑2) · 1))
148	48	recnd 8344	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘𝑤) ∈ ℂ)
149	148	sqcld 11087	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘𝑤)↑2) ∈ ℂ)
150	149	mulridd 8333	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (((abs‘𝑤)↑2) · 1) = ((abs‘𝑤)↑2))
151	147, 150	breqtrd 4151	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (((abs‘𝑤)↑2) · (3 / 4)) ≤ ((abs‘𝑤)↑2))
152	58, 65, 59, 131, 151	letrd 8440	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘𝑤) · (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1))) ≤ ((abs‘𝑤)↑2))
153	148	sqvald 11086	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘𝑤)↑2) = ((abs‘𝑤) · (abs‘𝑤)))
154	152, 153	breqtrd 4151	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘𝑤) · (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1))) ≤ ((abs‘𝑤) · (abs‘𝑤)))
155		absgt0ap 11843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℂ → (𝑤 # 0 ↔ 0 < (abs‘𝑤)))
156	155	ad2antrr 492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (𝑤 # 0 ↔ 0 < (abs‘𝑤)))
157	53, 156	mpbid 147	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → 0 < (abs‘𝑤))
158	48, 157	elrpd 10073	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘𝑤) ∈ ℝ⁺)
159	57, 48, 158	lemul2d 10121	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → ((abs‘((((exp‘𝑤) − 1) / 𝑤) − 1)) ≤ (abs‘𝑤) ↔ ((abs‘𝑤) · (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1))) ≤ ((abs‘𝑤) · (abs‘𝑤))))
160	154, 159	mpbird 167	. . . . . . . . . . . . 13 ⊢ (((𝑤 ∈ ℂ ∧ 𝑤 # 0) ∧ (abs‘𝑤) < 1) → (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1)) ≤ (abs‘𝑤))
161	160	ad2ant2l 512	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1)) ≤ (abs‘𝑤))
162		simprl 535	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (abs‘𝑤) < 𝑥)
163	42, 44, 46, 161, 162	lelttrd 8441	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (abs‘((((exp‘𝑤) − 1) / 𝑤) − 1)) < 𝑥)
164	39, 163	eqbrtrd 4147	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) ∧ ((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1)) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥)
165	164	ex 115	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → (((abs‘𝑤) < 𝑥 ∧ (abs‘𝑤) < 1) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥))
166	24, 165	sylbid 150	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → ((abs‘(𝑤 − 0)) < inf({𝑥, 1}, ℝ, < ) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥))
167	166	adantld 278	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑤 ∈ ℂ ∧ 𝑤 # 0)) → ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < inf({𝑥, 1}, ℝ, < )) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥))
168	13, 167	sylan2b 287	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0}) → ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < inf({𝑥, 1}, ℝ, < )) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥))
169	168	ralrimiva 2623	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ∀𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < inf({𝑥, 1}, ℝ, < )) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥))
170		brimralrspcev 4185	. . . . 5 ⊢ ((inf({𝑥, 1}, ℝ, < ) ∈ ℝ⁺ ∧ ∀𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < inf({𝑥, 1}, ℝ, < )) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥)) → ∃𝑦 ∈ ℝ⁺ ∀𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < 𝑦) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥))
171	11, 169, 170	syl2anc 415	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ∃𝑦 ∈ ℝ⁺ ∀𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < 𝑦) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥))
172	171	rgen 2603	. . 3 ⊢ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < 𝑦) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥)
173		elrabi 2979	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → 𝑧 ∈ ℂ)
174		efcl 12409	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (exp‘𝑧) ∈ ℂ)
175	173, 174	syl 14	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → (exp‘𝑧) ∈ ℂ)
176		1cnd 8332	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → 1 ∈ ℂ)
177	175, 176	subcld 8627	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → ((exp‘𝑧) − 1) ∈ ℂ)
178		breq1 4128	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝑢 # 0 ↔ 𝑧 # 0))
179	178	elrab 2982	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↔ (𝑧 ∈ ℂ ∧ 𝑧 # 0))
180	179	simprbi 275	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → 𝑧 # 0)
181	177, 173, 180	divclapd 9110	. . . . . . 7 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → (((exp‘𝑧) − 1) / 𝑧) ∈ ℂ)
182	25, 181	fmpti 5851	. . . . . 6 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)):{𝑢 ∈ ℂ ∣ 𝑢 # 0}⟶ℂ
183	182	a1i 9	. . . . 5 ⊢ (⊤ → (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)):{𝑢 ∈ ℂ ∣ 𝑢 # 0}⟶ℂ)
184		apsscn 8965	. . . . . 6 ⊢ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ⊆ ℂ
185	184	a1i 9	. . . . 5 ⊢ (⊤ → {𝑢 ∈ ℂ ∣ 𝑢 # 0} ⊆ ℂ)
186		0cnd 8309	. . . . 5 ⊢ (⊤ → 0 ∈ ℂ)
187	183, 185, 186	ellimc3ap 15685	. . . 4 ⊢ (⊤ → (1 ∈ ((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)) lim_ℂ 0) ↔ (1 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < 𝑦) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥))))
188	187	mptru 1411	. . 3 ⊢ (1 ∈ ((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)) lim_ℂ 0) ↔ (1 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑤 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ((𝑤 # 0 ∧ (abs‘(𝑤 − 0)) < 𝑦) → (abs‘(((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧))‘𝑤) − 1)) < 𝑥)))
189	8, 172, 188	mpbir2an 955	. 2 ⊢ 1 ∈ ((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)) lim_ℂ 0)
190	2	cntoptopon 15556	. . . . 5 ⊢ (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)
191	190	toponrestid 15045	. . . 4 ⊢ (MetOpen‘(abs ∘ − )) = ((MetOpen‘(abs ∘ − )) ↾_t ℂ)
192	173	subid1d 8616	. . . . . . 7 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → (𝑧 − 0) = 𝑧)
193	192	oveq2d 6091	. . . . . 6 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → (((exp‘𝑧) − (exp‘0)) / (𝑧 − 0)) = (((exp‘𝑧) − (exp‘0)) / 𝑧))
194		ef0 12417	. . . . . . . 8 ⊢ (exp‘0) = 1
195	194	oveq2i 6086	. . . . . . 7 ⊢ ((exp‘𝑧) − (exp‘0)) = ((exp‘𝑧) − 1)
196	195	oveq1i 6085	. . . . . 6 ⊢ (((exp‘𝑧) − (exp‘0)) / 𝑧) = (((exp‘𝑧) − 1) / 𝑧)
197	193, 196	eqtr2di 2288	. . . . 5 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} → (((exp‘𝑧) − 1) / 𝑧) = (((exp‘𝑧) − (exp‘0)) / (𝑧 − 0)))
198	197	mpteq2ia 4212	. . . 4 ⊢ (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)) = (𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − (exp‘0)) / (𝑧 − 0)))
199		ssidd 3269	. . . 4 ⊢ (⊤ → ℂ ⊆ ℂ)
200		eff 12408	. . . . 5 ⊢ exp:ℂ⟶ℂ
201	200	a1i 9	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
202	191, 2, 198, 199, 201, 199	eldvap 15706	. . 3 ⊢ (⊤ → (0(ℂ D exp)1 ↔ (0 ∈ ((int‘(MetOpen‘(abs ∘ − )))‘ℂ) ∧ 1 ∈ ((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)) lim_ℂ 0))))
203	202	mptru 1411	. 2 ⊢ (0(ℂ D exp)1 ↔ (0 ∈ ((int‘(MetOpen‘(abs ∘ − )))‘ℂ) ∧ 1 ∈ ((𝑧 ∈ {𝑢 ∈ ℂ ∣ 𝑢 # 0} ↦ (((exp‘𝑧) − 1) / 𝑧)) lim_ℂ 0)))
204	7, 189, 203	mpbir2an 955	1 ⊢ 0(ℂ D exp)1

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ⊤wtru 1403 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 {crab 2532 ⊆ wss 3220 {cpr 3706 class class class wbr 4125 ↦ cmpt 4187 ∘ ccom 4773 ⟶wf 5368 ‘cfv 5372 (class class class)co 6075 infcinf 7313 ℂcc 8167 ℝcr 8168 0cc0 8169 1c1 8170 + caddc 8172 · cmul 8174 < clt 8350 ≤ cle 8351 − cmin 8487 # cap 8899 / cdiv 8992 ℕcn 9283 2c2 9334 3c3 9335 4c4 9336 ℕ₀cn0 9542 ℤ_≥cuz 9900 ℝ⁺crp 10033 ↑cexp 10953 !cfa 11141 abscabs 11741 Σcsu 12097 expce 12387 MetOpencmopn 14850 Topctop 15021 intcnt 15117 lim_ℂ climc 15678 D cdv 15679

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 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 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem depends on definitions: df-bi 117 df-stab 843 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 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-oadd 6681 df-er 6797 df-map 6914 df-pm 6915 df-en 7013 df-dom 7014 df-fin 7015 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-xneg 10153 df-xadd 10154 df-ico 10275 df-fz 10391 df-fzo 10528 df-seqfrec 10863 df-exp 10954 df-fac 11142 df-ihash 11193 df-shft 11558 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-sumdc 12098 df-ef 12393 df-rest 13572 df-topgen 13591 df-psmet 14852 df-xmet 14853 df-met 14854 df-bl 14855 df-mopn 14856 df-top 15022 df-topon 15035 df-bases 15067 df-ntr 15120 df-limced 15680 df-dvap 15681

This theorem is referenced by: dvef 15751

W3C validator