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Theorem dvef 25489

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvfcn 25417	. . . . . . 7 ⊢ (ℂ D exp):dom (ℂ D exp)⟶ℂ
2		dvbsss 25411	. . . . . . . . 9 ⊢ dom (ℂ D exp) ⊆ ℂ
3		subcl 11456	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
4	3	ancoms 460	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
5		efadd 16034	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ (𝑧 − 𝑥) ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))))
6	4, 5	syldan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))))
7		pncan3 11465	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
8	7	fveq2d 6893	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = (exp‘𝑧))
9	6, 8	eqtr3d 2775	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))) = (exp‘𝑧))
10	9	mpteq2dva 5248	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧 − 𝑥)))) = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
11		cnex 11188	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
12	11	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ℂ ∈ V)
13		fvexd 6904	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ V)
14		fvexd 6904	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ V)
15		fconstmpt 5737	. . . . . . . . . . . . . . . 16 ⊢ (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
16	15	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
17		eqidd 2734	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
18	12, 13, 14, 16, 17	offval2 7687	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧 − 𝑥)))))
19		eff 16022	. . . . . . . . . . . . . . . 16 ⊢ exp:ℂ⟶ℂ
20	19	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → exp:ℂ⟶ℂ)
21	20	feqmptd 6958	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
22	10, 18, 21	3eqtr4d 2783	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = exp)
23	22	oveq2d 7422	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))) = (ℂ D exp))
24		efcl 16023	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
25		fconstg 6776	. . . . . . . . . . . . . . . 16 ⊢ ((exp‘𝑥) ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶{(exp‘𝑥)})
26	24, 25	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶{(exp‘𝑥)})
27	24	snssd 4812	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
28	26, 27	fssd 6733	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
29		ssidd 4005	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
30		efcl 16023	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 − 𝑥) ∈ ℂ → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
31	4, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ ℂ)
32	31	fmpttd 7112	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))):ℂ⟶ℂ)
33		0cnd 11204	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 0 ∈ ℂ)
34		1cnd 11206	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 1 ∈ ℂ)
35		c0ex 11205	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
36	35	snid 4664	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0}
37		opelxpi 5713	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
38	36, 37	mpan2 690	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
39		dvconst 25426	. . . . . . . . . . . . . . . . 17 ⊢ ((exp‘𝑥) ∈ ℂ → (ℂ D (ℂ × {(exp‘𝑥)})) = (ℂ × {0}))
40	24, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (ℂ D (ℂ × {(exp‘𝑥)})) = (ℂ × {0}))
41	38, 40	eleqtrrd 2837	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
42		df-br 5149	. . . . . . . . . . . . . . 15 ⊢ (𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0 ↔ ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
43	41, 42	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0)
44	20, 4	cofmpt 7127	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
45	44	oveq2d 7422	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))
46	4	fmpttd 7112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)):ℂ⟶ℂ)
47		oveq1 7413	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑥 → (𝑧 − 𝑥) = (𝑥 − 𝑥))
48		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))
49		ovex 7439	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 − 𝑥) ∈ V
50	47, 48, 49	fvmpt 6996	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
51		subid 11476	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) = 0)
52	50, 51	eqtrd 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
53		dveflem 25488	. . . . . . . . . . . . . . . . . 18 ⊢ 0(ℂ D exp)1
54	52, 53	eqbrtrdi 5187	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)(ℂ D exp)1)
55		1ex 11207	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ V
56	55	snid 4664	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ {1}
57		opelxpi 5713	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ {1}) → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
58	56, 57	mpan2 690	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
59		cnelprrecn 11200	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℂ ∈ {ℝ, ℂ}
60	59	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ → ℂ ∈ {ℝ, ℂ})
61		simpr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
62		1cnd 11206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
63	60	dvmptid 25466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1))
64		simpl 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
65		0cnd 11204	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
66		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
67	60, 66	dvmptc 25467	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
68	60, 61, 62, 63, 64, 65, 67	dvmptsub 25476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
69		1m0e1 12330	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 − 0) = 1
70	69	mpteq2i 5253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
71		fconstmpt 5737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
72	70, 71	eqtr4i 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
73	68, 72	eqtrdi 2789	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (ℂ × {1}))
74	58, 73	eleqtrrd 2837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
75		df-br 5149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))1 ↔ ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
76	74, 75	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))1)
77		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
78	20, 29, 46, 29, 29, 29, 34, 34, 54, 76, 77	dvcobr 25455	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))(1 · 1))
79		1t1e1 12371	. . . . . . . . . . . . . . . 16 ⊢ (1 · 1) = 1
80	78, 79	breqtrdi 5189	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))1)
81	45, 80	breqdi 5163	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))1)
82	28, 29, 32, 29, 29, 33, 34, 43, 81, 77	dvmulbr 25448	. . . . . . . . . . . . 13 ⊢ (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))))
83	32, 66	ffvelcdmd 7085	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥) ∈ ℂ)
84	83	mul02d 11409	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) = 0)
85		fvex 6902	. . . . . . . . . . . . . . . . . 18 ⊢ (exp‘𝑥) ∈ V
86	85	fvconst2 7202	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
87	86	oveq2d 7422	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
88	24	mullidd 11229	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
89	87, 88	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
90	84, 89	oveq12d 7424	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
91	24	addlidd 11412	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
92	90, 91	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
93	82, 92	breqtrd 5174	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))(exp‘𝑥))
94	23, 93	breqdi 5163	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
95		vex 3479	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
96	95, 85	breldm 5907	. . . . . . . . . . 11 ⊢ (𝑥(ℂ D exp)(exp‘𝑥) → 𝑥 ∈ dom (ℂ D exp))
97	94, 96	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
98	97	ssriv 3986	. . . . . . . . 9 ⊢ ℂ ⊆ dom (ℂ D exp)
99	2, 98	eqssi 3998	. . . . . . . 8 ⊢ dom (ℂ D exp) = ℂ
100	99	feq2i 6707	. . . . . . 7 ⊢ ((ℂ D exp):dom (ℂ D exp)⟶ℂ ↔ (ℂ D exp):ℂ⟶ℂ)
101	1, 100	mpbi 229	. . . . . 6 ⊢ (ℂ D exp):ℂ⟶ℂ
102	101	a1i 11	. . . . 5 ⊢ (⊤ → (ℂ D exp):ℂ⟶ℂ)
103	102	feqmptd 6958	. . . 4 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
104		ffun 6718	. . . . . . 7 ⊢ ((ℂ D exp):dom (ℂ D exp)⟶ℂ → Fun (ℂ D exp))
105	1, 104	ax-mp 5	. . . . . 6 ⊢ Fun (ℂ D exp)
106		funbrfv 6940	. . . . . 6 ⊢ (Fun (ℂ D exp) → (𝑥(ℂ D exp)(exp‘𝑥) → ((ℂ D exp)‘𝑥) = (exp‘𝑥)))
107	105, 94, 106	mpsyl 68	. . . . 5 ⊢ (𝑥 ∈ ℂ → ((ℂ D exp)‘𝑥) = (exp‘𝑥))
108	107	mpteq2ia 5251	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
109	103, 108	eqtrdi 2789	. . 3 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
110	19	a1i 11	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
111	110	feqmptd 6958	. . 3 ⊢ (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
112	109, 111	eqtr4d 2776	. 2 ⊢ (⊤ → (ℂ D exp) = exp)
113	112	mptru 1549	1 ⊢ (ℂ D exp) = exp

Colors of variables: wff setvar class

Syntax hints: ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3475 {csn 4628 {cpr 4630 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 dom cdm 5676 ∘ ccom 5680 Fun wfun 6535 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ∘_f cof 7665 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 − cmin 11441 expce 16002 TopOpenctopn 17364 ℂ_fldccnfld 20937 D cdv 25372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-haus 22811 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-xms 23818 df-ms 23819 df-tms 23820 df-cncf 24386 df-limc 25375 df-dv 25376

This theorem is referenced by: dvsincos 25490 efcn 25947 efcvx 25953 pige3ALT 26021 dvrelog 26137 dvlog 26151 dvcxp1 26238 dvcxp2 26239 dvcncxp1 26241 itgexpif 33607 dvsef 43077 expgrowthi 43078 expgrowth 43080

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