Theorem dvef 13328

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 7877	. . . . . . . 8 ⊢ ℂ ∈ V
2		eff 11604	. . . . . . . 8 ⊢ exp:ℂ⟶ℂ
3		fpmg 6640	. . . . . . . 8 ⊢ ((ℂ ∈ V ∧ ℂ ∈ V ∧ exp:ℂ⟶ℂ) → exp ∈ (ℂ ↑pm ℂ))
4	1, 1, 2, 3	mp3an 1327	. . . . . . 7 ⊢ exp ∈ (ℂ ↑pm ℂ)
5		dvfcnpm 13299	. . . . . . 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 5340	. . . . . 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 8097	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
10	9	ancoms 266	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
11		efadd 11616	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (𝑧 − 𝑥) ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))))
12	10, 11	syldan 280	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))))
13		pncan3 8106	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
14	13	fveq2d 5490	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = (exp‘𝑧))
15	12, 14	eqtr3d 2200	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))) = (exp‘𝑧))
16	15	mpteq2dva 4072	. . . . . . . 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 11605	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
19	18	adantr 274	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
20		efcl 11605	. . . . . . . . . 10 ⊢ ((𝑧 − 𝑥) ∈ ℂ → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
21	10, 20	syl 14	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
22		fconstmpt 4651	. . . . . . . . . 10 ⊢ (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
23	22	a1i 9	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
24		eqidd 2166	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
25	17, 19, 21, 23, 24	offval2 6065	. . . . . . . 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 5539	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
28	16, 25, 27	3eqtr4d 2208	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = exp)
29	28	oveq2d 5858	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))) = (ℂ D exp))
30		fconstg 5384	. . . . . . . . . 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 3718	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
33	31, 32	fssd 5350	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
34		ssidd 3163	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
35	21	fmpttd 5640	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))):ℂ⟶ℂ)
36		c0ex 7893	. . . . . . . . . . . 12 ⊢ 0 ∈ V
37	36	snid 3607	. . . . . . . . . . 11 ⊢ 0 ∈ {0}
38		opelxpi 4636	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
39	37, 38	mpan2 422	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
40		dvconst 13301	. . . . . . . . . . 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 2246	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
43		df-br 3983	. . . . . . . . 9 ⊢ (𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0 ↔ ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
44	42, 43	sylibr 133	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0)
45	26, 10	cofmpt 5654	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
46	45	oveq2d 5858	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))
47	10	fmpttd 5640	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)):ℂ⟶ℂ)
48		simpr 109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → 𝑢 ∈ ℂ)
49	48	adantr 274	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑢 ∈ ℂ)
50		simpl 108	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → 𝑥 ∈ ℂ)
51	50	adantr 274	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑥 ∈ ℂ)
52		simpr 109	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → 𝑢 # 𝑥)
53	49, 51, 52	subap0d 8542	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (𝑢 − 𝑥) # 0)
54		eqid 2165	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))
55		oveq1 5849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑢 → (𝑧 − 𝑥) = (𝑢 − 𝑥))
56		subcl 8097	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑢 − 𝑥) ∈ ℂ)
57	56	ancoms 266	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑢 − 𝑥) ∈ ℂ)
58	54, 55, 48, 57	fvmptd3 5579	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) = (𝑢 − 𝑥))
59		oveq1 5849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑧 − 𝑥) = (𝑥 − 𝑥))
60		id 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
61	60, 60	subcld 8209	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) ∈ ℂ)
62	61	adantr 274	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) ∈ ℂ)
63	54, 59, 50, 62	fvmptd3 5579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
64		subid 8117	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) = 0)
65	64	adantr 274	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑥 − 𝑥) = 0)
66	63, 65	eqtrd 2198	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
67	58, 66	breq12d 3995	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) ↔ (𝑢 − 𝑥) # 0))
68	67	adantr 274	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → (((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) ↔ (𝑢 − 𝑥) # 0))
69	53, 68	mpbird 166	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) ∧ 𝑢 # 𝑥) → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥))
70	69	ex 114	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)))
71	70	ralrimiva 2539	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ∀𝑢 ∈ ℂ (𝑢 # 𝑥 → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑢) # ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)))
72	54, 59, 60, 61	fvmptd3 5579	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
73	72, 64	eqtrd 2198	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
74		dveflem 13327	. . . . . . . . . . . 12 ⊢ 0(ℂ D exp)1
75	73, 74	eqbrtrdi 4021	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)(ℂ D exp)1)
76		1ex 7894	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
77	76	snid 3607	. . . . . . . . . . . . . 14 ⊢ 1 ∈ {1}
78		opelxpi 4636	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ {1}) → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
79	77, 78	mpan2 422	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
80		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
81		1cnd 7915	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
82		dvmptidcn 13318	. . . . . . . . . . . . . . . 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 108	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
85		0cnd 7892	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
86	60	dvmptccn 13319	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
87	80, 81, 83, 84, 85, 86	dvmptsubcn 13325	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
88		1m0e1 8970	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
89	88	mpteq2i 4069	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
90		fconstmpt 4651	. . . . . . . . . . . . . . 15 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
91	89, 90	eqtr4i 2189	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
92	87, 91	eqtrdi 2215	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (ℂ × {1}))
93	79, 92	eleqtrrd 2246	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
94		df-br 3983	. . . . . . . . . . . 12 ⊢ (𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))1 ↔ ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
95	93, 94	sylibr 133	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))1)
96		eqid 2165	. . . . . . . . . . 11 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
97	26, 34, 47, 34, 71, 34, 34, 75, 95, 96	dvcoapbr 13311	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))(1 · 1))
98		1t1e1 9009	. . . . . . . . . 10 ⊢ (1 · 1) = 1
99	97, 98	breqtrdi 4023	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))1)
100	46, 99	breqdi 3997	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))1)
101	33, 34, 35, 34, 44, 100, 96	dvmulxxbr 13306	. . . . . . 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	ffvelrnd 5621	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥) ∈ ℂ)
103	102	mul02d 8290	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) = 0)
104		fvconst2g 5699	. . . . . . . . . . . 12 ⊢ (((exp‘𝑥) ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
105	18, 104	mpancom 419	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
106	105	oveq2d 5858	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
107	18	mulid2d 7917	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
108	106, 107	eqtrd 2198	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
109	103, 108	oveq12d 5860	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
110	18	addid2d 8048	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
111	109, 110	eqtrd 2198	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
112	101, 111	breqtrd 4008	. . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘𝑓 · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))(exp‘𝑥))
113	29, 112	breqdi 3997	. . . . 5 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
114		funbrfv 5525	. . . . 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 4068	. . 3 ⊢ (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
117		ssid 3162	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
118		dvbsssg 13295	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ exp ∈ (ℂ ↑pm ℂ)) → dom (ℂ D exp) ⊆ ℂ)
119	117, 4, 118	mp2an 423	. . . . . . . 8 ⊢ dom (ℂ D exp) ⊆ ℂ
120		breldmg 4810	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ (exp‘𝑥) ∈ ℂ ∧ 𝑥(ℂ D exp)(exp‘𝑥)) → 𝑥 ∈ dom (ℂ D exp))
121	18, 113, 120	mpd3an23 1329	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
122	121	ssriv 3146	. . . . . . . 8 ⊢ ℂ ⊆ dom (ℂ D exp)
123	119, 122	eqssi 3158	. . . . . . 7 ⊢ dom (ℂ D exp) = ℂ
124	123	feq2i 5331	. . . . . 6 ⊢ ((ℂ D exp):dom (ℂ D exp)⟶ℂ ↔ (ℂ D exp):ℂ⟶ℂ)
125	6, 124	mpbi 144	. . . . 5 ⊢ (ℂ D exp):ℂ⟶ℂ
126	125	a1i 9	. . . 4 ⊢ (⊤ → (ℂ D exp):ℂ⟶ℂ)
127	126	feqmptd 5539	. . 3 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
128	2	a1i 9	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
129	128	feqmptd 5539	. . 3 ⊢ (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
130	116, 127, 129	3eqtr4a 2225	. 2 ⊢ (⊤ → (ℂ D exp) = exp)
131	130	mptru 1352	1 ⊢ (ℂ D exp) = exp

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ⊤wtru 1344   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  {csn 3576  ⟨cop 3579   class class class wbr 3982   ↦ cmpt 4043   × cxp 4602  dom cdm 4604   ∘ ccom 4608  Fun wfun 5182  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842   ∘_𝑓 cof 6048   ↑_pm cpm 6615  ℂcc 7751  0cc0 7753  1c1 7754   + caddc 7756   · cmul 7758   − cmin 8069   # cap 8479  abscabs 10939  expce 11583  MetOpencmopn 12625   D cdv 13264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by:  efcn  13329