Theorem dvef 15450

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 8155	. . . . . . . 8 ⊢ ℂ ∈ V
2		eff 12223	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
3		fpmg 6842	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ exp:ℂ⟶ℂ) → exp ∈ (ℂ ↑pm ℂ))
4	1, 1, 2, 3	mp3an 1373	. . . . . . 7 ⊢ exp ∈ (ℂ ↑pm ℂ)
5		dvfcnpm 15413	. . . . . . 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 5485	. . . . . 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 8377	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
10	9	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
11		efadd 12235	. . . . . . . . . . 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 8386	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
14	13	fveq2d 5643	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = (exp‘𝑧))
15	12, 14	eqtr3d 2266	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))) = (exp‘𝑧))
16	15	mpteq2dva 4179	. . . . . . . 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 12224	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
20		efcl 12224	. . . . . . . . . 10 ⊢ ((𝑧 − 𝑥) ∈ ℂ → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
21	10, 20	syl 14	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
22		fconstmpt 4773	. . . . . . . . . 10 ⊢ (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
23	22	a1i 9	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
24		eqidd 2232	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
25	17, 19, 21, 23, 24	offval2 6250	. . . . . . . 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 5699	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
28	16, 25, 27	3eqtr4d 2274	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = exp)
29	28	oveq2d 6033	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))) = (ℂ D exp))
30		fconstg 5533	. . . . . . . . . 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 3818	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
33	31, 32	fssd 5495	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
34		ssidd 3248	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
35	21	fmpttd 5802	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))):ℂ⟶ℂ)
36		c0ex 8172	. . . . . . . . . . . 12 ⊢ 0 ∈ V
37	36	snid 3700	. . . . . . . . . . 11 ⊢ 0 ∈ {0}
38		opelxpi 4757	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
39	37, 38	mpan2 425	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
40		dvconst 15417	. . . . . . . . . . 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 2311	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
43		df-br 4089	. . . . . . . . 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 5816	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
46	45	oveq2d 6033	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))
47	10	fmpttd 5802	. . . . . . . . . . 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 8823	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (𝑢 − 𝑥) # 0)
54		eqid 2231	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))
55		oveq1 6024	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑢 → (𝑧 − 𝑥) = (𝑢 − 𝑥))
56		subcl 8377	. . . . . . . . . . . . . . . . . 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 5740	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) = (𝑢 − 𝑥))
59		oveq1 6024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑧 − 𝑥) = (𝑥 − 𝑥))
60		id 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
61	60, 60	subcld 8489	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) ∈ ℂ)
62	61	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) ∈ ℂ)
63	54, 59, 50, 62	fvmptd3 5740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
64		subid 8397	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) = 0)
65	64	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) = 0)
66	63, 65	eqtrd 2264	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
67	58, 66	breq12d 4101	. . . . . . . . . . . . . . 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 2605	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ∀𝑢 ∈ ℂ (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)))
72	54, 59, 60, 61	fvmptd3 5740	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
73	72, 64	eqtrd 2264	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
74		dveflem 15449	. . . . . . . . . . . 12 ⊢ 0(ℂ D exp)1
75	73, 74	eqbrtrdi 4127	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)(ℂ D exp)1)
76		1ex 8173	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
77	76	snid 3700	. . . . . . . . . . . . . 14 ⊢ 1 ∈ {1}
78		opelxpi 4757	. . . . . . . . . . . . . 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 8194	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
82		dvmptidcn 15437	. . . . . . . . . . . . . . . 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 8171	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
86	60	dvmptccn 15438	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
87	80, 81, 83, 84, 85, 86	dvmptsubcn 15446	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
88		1m0e1 9255	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
89	88	mpteq2i 4176	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
90		fconstmpt 4773	. . . . . . . . . . . . . . 15 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
91	89, 90	eqtr4i 2255	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
92	87, 91	eqtrdi 2280	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (ℂ × {1}))
93	79, 92	eleqtrrd 2311	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
94		df-br 4089	. . . . . . . . . . . 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 2231	. . . . . . . . . . 11 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
97	26, 34, 47, 34, 71, 34, 34, 75, 95, 96	dvcoapbr 15430	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))(1 · 1))
98		1t1e1 9295	. . . . . . . . . 10 ⊢ (1 · 1) = 1
99	97, 98	breqtrdi 4129	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))1)
100	46, 99	breqdi 4103	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))1)
101	33, 34, 35, 34, 44, 100, 96	dvmulxxbr 15425	. . . . . . 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 5783	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥) ∈ ℂ)
103	102	mul02d 8570	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) = 0)
104		fvconst2g 5867	. . . . . . . . . . . 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 6033	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
107	18	mulid2d 8197	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
108	106, 107	eqtrd 2264	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
109	103, 108	oveq12d 6035	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
110	18	addlidd 8328	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
111	109, 110	eqtrd 2264	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
112	101, 111	breqtrd 4114	. . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))(exp‘𝑥))
113	29, 112	breqdi 4103	. . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
114		funbrfv 5682	. . . . 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 4175	. . 3 ⊢ (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
117		ssid 3247	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
118		dvbsssg 15409	. . . . . . . . 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 4937	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (exp‘𝑥) ∈ ℂ ∧ 𝑥(ℂ D exp)(exp‘𝑥)) → 𝑥 ∈ dom (ℂ D exp))
121	18, 113, 120	mpd3an23 1375	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
122	121	ssriv 3231	. . . . . . . 8 ⊢ ℂ ⊆ dom (ℂ D exp)
123	119, 122	eqssi 3243	. . . . . . 7 ⊢ dom (ℂ D exp) = ℂ
124	123	feq2i 5476	. . . . . 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 5699	. . 3 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
128	2	a1i 9	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
129	128	feqmptd 5699	. . 3 ⊢ (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
130	116, 127, 129	3eqtr4a 2290	. 2 ⊢ (⊤ → (ℂ D exp) = exp)
131	130	mptru 1406	1 ⊢ (ℂ D exp) = exp

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397  ⊤wtru 1398   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  {csn 3669  ⟨cop 3672   class class class wbr 4088   ↦ cmpt 4150   × cxp 4723  dom cdm 4725   ∘ ccom 4729  Fun wfun 5320  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017   ∘_𝑓 cof 6232   ↑_pm cpm 6817  ℂcc 8029  0cc0 8031  1c1 8032   + caddc 8034   · cmul 8036   − cmin 8349   # cap 8760  abscabs 11557  expce 12202  MetOpencmopn 14554   D cdv 15378

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151  ax-addf 8153  ax-mulf 8154

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-of 6234  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-map 6818  df-pm 6819  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-xneg 10006  df-xadd 10007  df-ico 10128  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-fac 10987  df-bc 11009  df-ihash 11037  df-shft 11375  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914  df-ef 12208  df-rest 13323  df-topgen 13342  df-psmet 14556  df-xmet 14557  df-met 14558  df-bl 14559  df-mopn 14560  df-top 14721  df-topon 14734  df-bases 14766  df-ntr 14819  df-cn 14911  df-cnp 14912  df-tx 14976  df-cncf 15294  df-limced 15379  df-dvap 15380

This theorem is referenced by:  efcn  15491