Theorem dvef 14873

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 7996	. . . . . . . 8 ⊢ ℂ ∈ V
2		eff 11806	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
3		fpmg 6728	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ exp:ℂ⟶ℂ) → exp ∈ (ℂ ↑pm ℂ))
4	1, 1, 2, 3	mp3an 1348	. . . . . . 7 ⊢ exp ∈ (ℂ ↑pm ℂ)
5		dvfcnpm 14844	. . . . . . 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 5406	. . . . . 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 8218	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
10	9	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
11		efadd 11818	. . . . . . . . . . 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 8227	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
14	13	fveq2d 5558	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = (exp‘𝑧))
15	12, 14	eqtr3d 2228	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))) = (exp‘𝑧))
16	15	mpteq2dva 4119	. . . . . . . 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 11807	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
20		efcl 11807	. . . . . . . . . 10 ⊢ ((𝑧 − 𝑥) ∈ ℂ → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
21	10, 20	syl 14	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
22		fconstmpt 4706	. . . . . . . . . 10 ⊢ (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
23	22	a1i 9	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
24		eqidd 2194	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
25	17, 19, 21, 23, 24	offval2 6146	. . . . . . . 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 5610	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
28	16, 25, 27	3eqtr4d 2236	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = exp)
29	28	oveq2d 5934	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))) = (ℂ D exp))
30		fconstg 5450	. . . . . . . . . 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 3763	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
33	31, 32	fssd 5416	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
34		ssidd 3200	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
35	21	fmpttd 5713	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))):ℂ⟶ℂ)
36		c0ex 8013	. . . . . . . . . . . 12 ⊢ 0 ∈ V
37	36	snid 3649	. . . . . . . . . . 11 ⊢ 0 ∈ {0}
38		opelxpi 4691	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
39	37, 38	mpan2 425	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
40		dvconst 14846	. . . . . . . . . . 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 2273	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
43		df-br 4030	. . . . . . . . 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 5727	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
46	45	oveq2d 5934	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))
47	10	fmpttd 5713	. . . . . . . . . . 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 8663	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (𝑢 − 𝑥) # 0)
54		eqid 2193	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))
55		oveq1 5925	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑢 → (𝑧 − 𝑥) = (𝑢 − 𝑥))
56		subcl 8218	. . . . . . . . . . . . . . . . . 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 5651	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) = (𝑢 − 𝑥))
59		oveq1 5925	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑧 − 𝑥) = (𝑥 − 𝑥))
60		id 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
61	60, 60	subcld 8330	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) ∈ ℂ)
62	61	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) ∈ ℂ)
63	54, 59, 50, 62	fvmptd3 5651	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
64		subid 8238	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) = 0)
65	64	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) = 0)
66	63, 65	eqtrd 2226	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
67	58, 66	breq12d 4042	. . . . . . . . . . . . . . 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 2567	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ∀𝑢 ∈ ℂ (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)))
72	54, 59, 60, 61	fvmptd3 5651	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
73	72, 64	eqtrd 2226	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
74		dveflem 14872	. . . . . . . . . . . 12 ⊢ 0(ℂ D exp)1
75	73, 74	eqbrtrdi 4068	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)(ℂ D exp)1)
76		1ex 8014	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
77	76	snid 3649	. . . . . . . . . . . . . 14 ⊢ 1 ∈ {1}
78		opelxpi 4691	. . . . . . . . . . . . . 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 8035	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
82		dvmptidcn 14863	. . . . . . . . . . . . . . . 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 8012	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
86	60	dvmptccn 14864	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
87	80, 81, 83, 84, 85, 86	dvmptsubcn 14870	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
88		1m0e1 9095	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
89	88	mpteq2i 4116	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
90		fconstmpt 4706	. . . . . . . . . . . . . . 15 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
91	89, 90	eqtr4i 2217	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
92	87, 91	eqtrdi 2242	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (ℂ × {1}))
93	79, 92	eleqtrrd 2273	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
94		df-br 4030	. . . . . . . . . . . 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 2193	. . . . . . . . . . 11 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
97	26, 34, 47, 34, 71, 34, 34, 75, 95, 96	dvcoapbr 14856	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))(1 · 1))
98		1t1e1 9134	. . . . . . . . . 10 ⊢ (1 · 1) = 1
99	97, 98	breqtrdi 4070	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))1)
100	46, 99	breqdi 4044	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))1)
101	33, 34, 35, 34, 44, 100, 96	dvmulxxbr 14851	. . . . . . 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 5694	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥) ∈ ℂ)
103	102	mul02d 8411	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) = 0)
104		fvconst2g 5772	. . . . . . . . . . . 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 5934	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
107	18	mulid2d 8038	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
108	106, 107	eqtrd 2226	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
109	103, 108	oveq12d 5936	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
110	18	addlidd 8169	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
111	109, 110	eqtrd 2226	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
112	101, 111	breqtrd 4055	. . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))(exp‘𝑥))
113	29, 112	breqdi 4044	. . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
114		funbrfv 5595	. . . . 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 4115	. . 3 ⊢ (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
117		ssid 3199	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
118		dvbsssg 14840	. . . . . . . . 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 4868	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (exp‘𝑥) ∈ ℂ ∧ 𝑥(ℂ D exp)(exp‘𝑥)) → 𝑥 ∈ dom (ℂ D exp))
121	18, 113, 120	mpd3an23 1350	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
122	121	ssriv 3183	. . . . . . . 8 ⊢ ℂ ⊆ dom (ℂ D exp)
123	119, 122	eqssi 3195	. . . . . . 7 ⊢ dom (ℂ D exp) = ℂ
124	123	feq2i 5397	. . . . . 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 5610	. . 3 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
128	2	a1i 9	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
129	128	feqmptd 5610	. . 3 ⊢ (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
130	116, 127, 129	3eqtr4a 2252	. 2 ⊢ (⊤ → (ℂ D exp) = exp)
131	130	mptru 1373	1 ⊢ (ℂ D exp) = exp

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ⊤wtru 1365   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3153  {csn 3618  ⟨cop 3621   class class class wbr 4029   ↦ cmpt 4090   × cxp 4657  dom cdm 4659   ∘ ccom 4663  Fun wfun 5248  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918   ∘_𝑓 cof 6128   ↑_pm cpm 6703  ℂcc 7870  0cc0 7872  1c1 7873   + caddc 7875   · cmul 7877   − cmin 8190   # cap 8600  abscabs 11141  expce 11785  MetOpencmopn 14037   D cdv 14809

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992  ax-addf 7994  ax-mulf 7995

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-disj 4007  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-map 6704  df-pm 6705  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-xneg 9838  df-xadd 9839  df-ico 9960  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-bc 10819  df-ihash 10847  df-shft 10959  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497  df-ef 11791  df-rest 12852  df-topgen 12871  df-psmet 14039  df-xmet 14040  df-met 14041  df-bl 14042  df-mopn 14043  df-top 14166  df-topon 14179  df-bases 14211  df-ntr 14264  df-cn 14356  df-cnp 14357  df-tx 14421  df-cncf 14726  df-limced 14810  df-dvap 14811

This theorem is referenced by:  efcn  14903