Theorem dvef 15718

Description: Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)

Assertion

Ref	Expression
dvef	⊢ (ℂ D exp) = exp

Proof of Theorem dvef

Dummy variables 𝑥 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 8267	. . . . . . . 8 ⊢ ℂ ∈ V
2		eff 12374	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
3		fpmg 6921	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ exp:ℂ⟶ℂ) → exp ∈ (ℂ ↑pm ℂ))
4	1, 1, 2, 3	mp3an 1374	. . . . . . 7 ⊢ exp ∈ (ℂ ↑pm ℂ)
5		dvfcnpm 15681	. . . . . . 7 ⊢ (exp ∈ (ℂ ↑pm ℂ) → (ℂ D exp):dom (ℂ D exp)⟶ℂ)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ (ℂ D exp):dom (ℂ D exp)⟶ℂ
7		ffun 5516	. . . . . 6 ⊢ ((ℂ D exp):dom (ℂ D exp)⟶ℂ → Fun (ℂ D exp))
8	6, 7	ax-mp 5	. . . . 5 ⊢ Fun (ℂ D exp)
9		subcl 8488	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
10	9	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
11		efadd 12386	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑧 − 𝑥) ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))))
12	10, 11	syldan 282	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))))
13		pncan3 8497	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
14	13	fveq2d 5679	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = (exp‘𝑧))
15	12, 14	eqtr3d 2269	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))) = (exp‘𝑧))
16	15	mpteq2dva 4205	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧 − 𝑥)))) = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
17	1	a1i 9	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ℂ ∈ V)
18		efcl 12375	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
20		efcl 12375	. . . . . . . . . 10 ⊢ ((𝑧 − 𝑥) ∈ ℂ → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
21	10, 20	syl 14	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
22		fconstmpt 4802	. . . . . . . . . 10 ⊢ (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
23	22	a1i 9	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
24		eqidd 2235	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
25	17, 19, 21, 23, 24	offval2 6291	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧 − 𝑥)))))
26	2	a1i 9	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → exp:ℂ⟶ℂ)
27	26	feqmptd 5735	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
28	16, 25, 27	3eqtr4d 2277	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = exp)
29	28	oveq2d 6074	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))) = (ℂ D exp))
30		fconstg 5569	. . . . . . . . . 10 ⊢ ((exp‘𝑥) ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶{(exp‘𝑥)})
31	18, 30	syl 14	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶{(exp‘𝑥)})
32	18	snssd 3844	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
33	31, 32	fssd 5527	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
34		ssidd 3263	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
35	21	fmpttd 5837	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))):ℂ⟶ℂ)
36		c0ex 8284	. . . . . . . . . . . 12 ⊢ 0 ∈ V
37	36	snid 3725	. . . . . . . . . . 11 ⊢ 0 ∈ {0}
38		opelxpi 4786	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
39	37, 38	mpan2 425	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
40		dvconst 15685	. . . . . . . . . . 11 ⊢ ((exp‘𝑥) ∈ ℂ → (ℂ D (ℂ × {(exp‘𝑥)})) = (ℂ × {0}))
41	18, 40	syl 14	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (ℂ D (ℂ × {(exp‘𝑥)})) = (ℂ × {0}))
42	39, 41	eleqtrrd 2314	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
43		df-br 4115	. . . . . . . . 9 ⊢ (𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0 ↔ ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
44	42, 43	sylibr 134	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0)
45	26, 10	cofmpt 5851	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
46	45	oveq2d 6074	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))
47	10	fmpttd 5837	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)):ℂ⟶ℂ)
48		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → 𝑢 ∈ ℂ)
49	48	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑢 ∈ ℂ)
50		simpl 109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → 𝑥 ∈ ℂ)
51	50	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑥 ∈ ℂ)
52		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑢 # 𝑥)
53	49, 51, 52	subap0d 8935	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (𝑢 − 𝑥) # 0)
54		eqid 2234	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))
55		oveq1 6065	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑢 → (𝑧 − 𝑥) = (𝑢 − 𝑥))
56		subcl 8488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑢 − 𝑥) ∈ ℂ)
57	56	ancoms 268	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑢 − 𝑥) ∈ ℂ)
58	54, 55, 48, 57	fvmptd3 5776	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) = (𝑢 − 𝑥))
59		oveq1 6065	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑧 − 𝑥) = (𝑥 − 𝑥))
60		id 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
61	60, 60	subcld 8600	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) ∈ ℂ)
62	61	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) ∈ ℂ)
63	54, 59, 50, 62	fvmptd3 5776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
64		subid 8508	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) = 0)
65	64	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) = 0)
66	63, 65	eqtrd 2267	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
67	58, 66	breq12d 4127	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) ↔ (𝑢 − 𝑥) # 0))
68	67	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) ↔ (𝑢 − 𝑥) # 0))
69	53, 68	mpbird 167	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥))
70	69	ex 115	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)))
71	70	ralrimiva 2617	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ∀𝑢 ∈ ℂ (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)))
72	54, 59, 60, 61	fvmptd3 5776	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
73	72, 64	eqtrd 2267	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
74		dveflem 15717	. . . . . . . . . . . 12 ⊢ 0(ℂ D exp)1
75	73, 74	eqbrtrdi 4153	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)(ℂ D exp)1)
76		1ex 8285	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
77	76	snid 3725	. . . . . . . . . . . . . 14 ⊢ 1 ∈ {1}
78		opelxpi 4786	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ {1}) → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
79	77, 78	mpan2 425	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
80		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
81		1cnd 8306	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
82		dvmptidcn 15705	. . . . . . . . . . . . . . . 16 ⊢ (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1)
83	82	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1))
84		simpl 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
85		0cnd 8283	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
86	60	dvmptccn 15706	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
87	80, 81, 83, 84, 85, 86	dvmptsubcn 15714	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
88		1m0e1 9367	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
89	88	mpteq2i 4202	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
90		fconstmpt 4802	. . . . . . . . . . . . . . 15 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
91	89, 90	eqtr4i 2258	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
92	87, 91	eqtrdi 2283	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (ℂ × {1}))
93	79, 92	eleqtrrd 2314	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
94		df-br 4115	. . . . . . . . . . . 12 ⊢ (𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))1 ↔ ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
95	93, 94	sylibr 134	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))1)
96		eqid 2234	. . . . . . . . . . 11 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
97	26, 34, 47, 34, 71, 34, 34, 75, 95, 96	dvcoapbr 15698	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))(1 · 1))
98		1t1e1 9407	. . . . . . . . . 10 ⊢ (1 · 1) = 1
99	97, 98	breqtrdi 4155	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))1)
100	46, 99	breqdi 4129	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))1)
101	33, 34, 35, 34, 44, 100, 96	dvmulxxbr 15693	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))))
102	35, 60	ffvelcdmd 5818	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥) ∈ ℂ)
103	102	mul02d 8682	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) = 0)
104		fvconst2g 5903	. . . . . . . . . . . 12 ⊢ (((exp‘𝑥) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
105	18, 104	mpancom 422	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
106	105	oveq2d 6074	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
107	18	mullidd 8308	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
108	106, 107	eqtrd 2267	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
109	103, 108	oveq12d 6076	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
110	18	addlidd 8439	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
111	109, 110	eqtrd 2267	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
112	101, 111	breqtrd 4140	. . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))(exp‘𝑥))
113	29, 112	breqdi 4129	. . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
114		funbrfv 5718	. . . . 5 ⊢ (Fun (ℂ D exp) → (𝑥(ℂ D exp)(exp‘𝑥) → ((ℂ D exp)‘𝑥) = (exp‘𝑥)))
115	8, 113, 114	mpsyl 65	. . . 4 ⊢ (𝑥 ∈ ℂ → ((ℂ D exp)‘𝑥) = (exp‘𝑥))
116	115	mpteq2ia 4201	. . 3 ⊢ (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
117		ssid 3262	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
118		dvbsssg 15677	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ exp ∈ (ℂ ↑pm ℂ)) → dom (ℂ D exp) ⊆ ℂ)
119	117, 4, 118	mp2an 426	. . . . . . . 8 ⊢ dom (ℂ D exp) ⊆ ℂ
120		breldmg 4967	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (exp‘𝑥) ∈ ℂ ∧ 𝑥(ℂ D exp)(exp‘𝑥)) → 𝑥 ∈ dom (ℂ D exp))
121	18, 113, 120	mpd3an23 1376	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
122	121	ssriv 3246	. . . . . . . 8 ⊢ ℂ ⊆ dom (ℂ D exp)
123	119, 122	eqssi 3258	. . . . . . 7 ⊢ dom (ℂ D exp) = ℂ
124	123	feq2i 5507	. . . . . 6 ⊢ ((ℂ D exp):dom (ℂ D exp)⟶ℂ ↔ (ℂ D exp):ℂ⟶ℂ)
125	6, 124	mpbi 145	. . . . 5 ⊢ (ℂ D exp):ℂ⟶ℂ
126	125	a1i 9	. . . 4 ⊢ (⊤ → (ℂ D exp):ℂ⟶ℂ)
127	126	feqmptd 5735	. . 3 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
128	2	a1i 9	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
129	128	feqmptd 5735	. . 3 ⊢ (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
130	116, 127, 129	3eqtr4a 2293	. 2 ⊢ (⊤ → (ℂ D exp) = exp)
131	130	mptru 1407	1 ⊢ (ℂ D exp) = exp

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ⊤wtru 1399   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214  {csn 3694  ⟨cop 3697   class class class wbr 4114   ↦ cmpt 4176   × cxp 4752  dom cdm 4754   ∘ ccom 4758  Fun wfun 5351  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058   ∘_𝑓 cof 6273   ↑_pm cpm 6896  ℂcc 8141  0cc0 8143  1c1 8144   + caddc 8146   · cmul 8148   − cmin 8460   # cap 8872  abscabs 11707  expce 12353  MetOpencmopn 14815   D cdv 15646

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263  ax-addf 8265  ax-mulf 8266

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-disj 4091  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-map 6897  df-pm 6898  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-ico 10246  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-bc 11135  df-ihash 11164  df-shft 11525  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-ef 12359  df-rest 13538  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-bases 15034  df-ntr 15087  df-cn 15179  df-cnp 15180  df-tx 15244  df-cncf 15562  df-limced 15647  df-dvap 15648

This theorem is referenced by:  efcn  15759