Theorem dvef 14191

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 7935	. . . . . . . 8 ⊢ ℂ ∈ V
2		eff 11671	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
3		fpmg 6674	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ exp:ℂ⟶ℂ) → exp ∈ (ℂ ↑pm ℂ))
4	1, 1, 2, 3	mp3an 1337	. . . . . . 7 ⊢ exp ∈ (ℂ ↑pm ℂ)
5		dvfcnpm 14162	. . . . . . 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 5369	. . . . . 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 8156	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
10	9	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
11		efadd 11683	. . . . . . . . . . 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 8165	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
14	13	fveq2d 5520	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = (exp‘𝑧))
15	12, 14	eqtr3d 2212	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))) = (exp‘𝑧))
16	15	mpteq2dva 4094	. . . . . . . 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 11672	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
20		efcl 11672	. . . . . . . . . 10 ⊢ ((𝑧 − 𝑥) ∈ ℂ → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
21	10, 20	syl 14	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
22		fconstmpt 4674	. . . . . . . . . 10 ⊢ (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
23	22	a1i 9	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
24		eqidd 2178	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
25	17, 19, 21, 23, 24	offval2 6098	. . . . . . . 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 5570	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
28	16, 25, 27	3eqtr4d 2220	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = exp)
29	28	oveq2d 5891	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))) = (ℂ D exp))
30		fconstg 5413	. . . . . . . . . 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 3738	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
33	31, 32	fssd 5379	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
34		ssidd 3177	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
35	21	fmpttd 5672	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))):ℂ⟶ℂ)
36		c0ex 7951	. . . . . . . . . . . 12 ⊢ 0 ∈ V
37	36	snid 3624	. . . . . . . . . . 11 ⊢ 0 ∈ {0}
38		opelxpi 4659	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
39	37, 38	mpan2 425	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
40		dvconst 14164	. . . . . . . . . . 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 2257	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
43		df-br 4005	. . . . . . . . 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 5686	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
46	45	oveq2d 5891	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))
47	10	fmpttd 5672	. . . . . . . . . . 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 8601	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (𝑢 − 𝑥) # 0)
54		eqid 2177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))
55		oveq1 5882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑢 → (𝑧 − 𝑥) = (𝑢 − 𝑥))
56		subcl 8156	. . . . . . . . . . . . . . . . . 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 5610	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) = (𝑢 − 𝑥))
59		oveq1 5882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑧 − 𝑥) = (𝑥 − 𝑥))
60		id 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
61	60, 60	subcld 8268	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) ∈ ℂ)
62	61	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) ∈ ℂ)
63	54, 59, 50, 62	fvmptd3 5610	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
64		subid 8176	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) = 0)
65	64	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) = 0)
66	63, 65	eqtrd 2210	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
67	58, 66	breq12d 4017	. . . . . . . . . . . . . . 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 2550	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ∀𝑢 ∈ ℂ (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)))
72	54, 59, 60, 61	fvmptd3 5610	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
73	72, 64	eqtrd 2210	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
74		dveflem 14190	. . . . . . . . . . . 12 ⊢ 0(ℂ D exp)1
75	73, 74	eqbrtrdi 4043	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)(ℂ D exp)1)
76		1ex 7952	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
77	76	snid 3624	. . . . . . . . . . . . . 14 ⊢ 1 ∈ {1}
78		opelxpi 4659	. . . . . . . . . . . . . 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 7973	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
82		dvmptidcn 14181	. . . . . . . . . . . . . . . 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 7950	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
86	60	dvmptccn 14182	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
87	80, 81, 83, 84, 85, 86	dvmptsubcn 14188	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
88		1m0e1 9032	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
89	88	mpteq2i 4091	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
90		fconstmpt 4674	. . . . . . . . . . . . . . 15 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
91	89, 90	eqtr4i 2201	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
92	87, 91	eqtrdi 2226	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (ℂ × {1}))
93	79, 92	eleqtrrd 2257	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
94		df-br 4005	. . . . . . . . . . . 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 2177	. . . . . . . . . . 11 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
97	26, 34, 47, 34, 71, 34, 34, 75, 95, 96	dvcoapbr 14174	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))(1 · 1))
98		1t1e1 9071	. . . . . . . . . 10 ⊢ (1 · 1) = 1
99	97, 98	breqtrdi 4045	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))1)
100	46, 99	breqdi 4019	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))1)
101	33, 34, 35, 34, 44, 100, 96	dvmulxxbr 14169	. . . . . . 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 5653	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥) ∈ ℂ)
103	102	mul02d 8349	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) = 0)
104		fvconst2g 5731	. . . . . . . . . . . 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 5891	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
107	18	mulid2d 7976	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
108	106, 107	eqtrd 2210	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
109	103, 108	oveq12d 5893	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
110	18	addid2d 8107	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
111	109, 110	eqtrd 2210	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
112	101, 111	breqtrd 4030	. . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))(exp‘𝑥))
113	29, 112	breqdi 4019	. . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
114		funbrfv 5555	. . . . 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 4090	. . 3 ⊢ (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
117		ssid 3176	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
118		dvbsssg 14158	. . . . . . . . 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 4834	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (exp‘𝑥) ∈ ℂ ∧ 𝑥(ℂ D exp)(exp‘𝑥)) → 𝑥 ∈ dom (ℂ D exp))
121	18, 113, 120	mpd3an23 1339	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
122	121	ssriv 3160	. . . . . . . 8 ⊢ ℂ ⊆ dom (ℂ D exp)
123	119, 122	eqssi 3172	. . . . . . 7 ⊢ dom (ℂ D exp) = ℂ
124	123	feq2i 5360	. . . . . 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 5570	. . 3 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
128	2	a1i 9	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
129	128	feqmptd 5570	. . 3 ⊢ (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
130	116, 127, 129	3eqtr4a 2236	. 2 ⊢ (⊤ → (ℂ D exp) = exp)
131	130	mptru 1362	1 ⊢ (ℂ D exp) = exp

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ⊤wtru 1354   ∈ wcel 2148  Vcvv 2738   ⊆ wss 3130  {csn 3593  ⟨cop 3596   class class class wbr 4004   ↦ cmpt 4065   × cxp 4625  dom cdm 4627   ∘ ccom 4631  Fun wfun 5211  ⟶wf 5213  ‘cfv 5217  (class class class)co 5875   ∘_𝑓 cof 6081   ↑_pm cpm 6649  ℂcc 7809  0cc0 7811  1c1 7812   + caddc 7814   · cmul 7816   − cmin 8128   # cap 8538  abscabs 11006  expce 11650  MetOpencmopn 13448   D cdv 14127

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931  ax-addf 7933  ax-mulf 7934

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-disj 3982  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-of 6083  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-map 6650  df-pm 6651  df-en 6741  df-dom 6742  df-fin 6743  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-xneg 9772  df-xadd 9773  df-ico 9894  df-fz 10009  df-fzo 10143  df-seqfrec 10446  df-exp 10520  df-fac 10706  df-bc 10728  df-ihash 10756  df-shft 10824  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287  df-sumdc 11362  df-ef 11656  df-rest 12690  df-topgen 12709  df-psmet 13450  df-xmet 13451  df-met 13452  df-bl 13453  df-mopn 13454  df-top 13501  df-topon 13514  df-bases 13546  df-ntr 13599  df-cn 13691  df-cnp 13692  df-tx 13756  df-cncf 14061  df-limced 14128  df-dvap 14129

This theorem is referenced by:  efcn  14192