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Theorem dvef 25940

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 25865	. . . . . . 7 ⊢ (ℂ D exp):dom (ℂ D exp)⟶ℂ
2		dvbsss 25859	. . . . . . . . 9 ⊢ dom (ℂ D exp) ⊆ ℂ
3		subcl 11379	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
4	3	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
5		efadd 16017	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ (𝑧 − 𝑥) ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))))
6	4, 5	syldan 591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))))
7		pncan3 11388	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
8	7	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = (exp‘𝑧))
9	6, 8	eqtr3d 2773	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((exp‘𝑥) · (exp‘(𝑧 − 𝑥))) = (exp‘𝑧))
10	9	mpteq2dva 5191	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧 − 𝑥)))) = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
11		cnex 11107	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
12	11	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ℂ ∈ V)
13		fvexd 6849	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ V)
14		fvexd 6849	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ V)
15		fconstmpt 5686	. . . . . . . . . . . . . . . 16 ⊢ (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
16	15	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
17		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
18	12, 13, 14, 16, 17	offval2 7642	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧 − 𝑥)))))
19		eff 16004	. . . . . . . . . . . . . . . 16 ⊢ exp:ℂ⟶ℂ
20	19	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → exp:ℂ⟶ℂ)
21	20	feqmptd 6902	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
22	10, 18, 21	3eqtr4d 2781	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = exp)
23	22	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))) = (ℂ D exp))
24		efcl 16005	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
25		fconstg 6721	. . . . . . . . . . . . . . . 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 4765	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
28	26, 27	fssd 6679	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
29		ssidd 3957	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
30		efcl 16005	. . . . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))):ℂ⟶ℂ)
33		c0ex 11126	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
34	33	snid 4619	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0}
35		opelxpi 5661	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
36	34, 35	mpan2 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
37		dvconst 25874	. . . . . . . . . . . . . . . . 17 ⊢ ((exp‘𝑥) ∈ ℂ → (ℂ D (ℂ × {(exp‘𝑥)})) = (ℂ × {0}))
38	24, 37	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (ℂ D (ℂ × {(exp‘𝑥)})) = (ℂ × {0}))
39	36, 38	eleqtrrd 2839	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
40		df-br 5099	. . . . . . . . . . . . . . 15 ⊢ (𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0 ↔ ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
41	39, 40	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (ℂ × {(exp‘𝑥)}))0)
42	20, 4	cofmpt 7077	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
43	42	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))
44	4	fmpttd 7060	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)):ℂ⟶ℂ)
45		oveq1 7365	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑥 → (𝑧 − 𝑥) = (𝑥 − 𝑥))
46		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))
47		ovex 7391	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 − 𝑥) ∈ V
48	45, 46, 47	fvmpt 6941	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
49		subid 11400	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) = 0)
50	48, 49	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
51		dveflem 25939	. . . . . . . . . . . . . . . . . 18 ⊢ 0(ℂ D exp)1
52	50, 51	eqbrtrdi 5137	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)(ℂ D exp)1)
53		1ex 11128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ V
54	53	snid 4619	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ {1}
55		opelxpi 5661	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ {1}) → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
56	54, 55	mpan2 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
57		cnelprrecn 11119	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℂ ∈ {ℝ, ℂ}
58	57	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ → ℂ ∈ {ℝ, ℂ})
59		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ)
60		1cnd 11127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
61	58	dvmptid 25917	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1))
62		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
63		0cnd 11125	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
64		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
65	58, 64	dvmptc 25918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
66	58, 59, 60, 61, 62, 63, 65	dvmptsub 25927	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
67		1m0e1 12261	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 − 0) = 1
68	67	mpteq2i 5194	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
69		fconstmpt 5686	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
70	68, 69	eqtr4i 2762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
71	66, 70	eqtrdi 2787	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (ℂ × {1}))
72	56, 71	eleqtrrd 2839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
73		df-br 5099	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))1 ↔ ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
74	72, 73	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))1)
75		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
76	20, 29, 44, 29, 29, 29, 52, 74, 75	dvcobr 25905	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))(1 · 1))
77		1t1e1 12302	. . . . . . . . . . . . . . . 16 ⊢ (1 · 1) = 1
78	76, 77	breqtrdi 5139	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))1)
79	43, 78	breqdi 5113	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))1)
80	28, 29, 32, 29, 29, 41, 79, 75	dvmulbr 25897	. . . . . . . . . . . . 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))))
81	32, 64	ffvelcdmd 7030	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥) ∈ ℂ)
82	81	mul02d 11331	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) = 0)
83		fvex 6847	. . . . . . . . . . . . . . . . . 18 ⊢ (exp‘𝑥) ∈ V
84	83	fvconst2 7150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
85	84	oveq2d 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
86	24	mullidd 11150	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
87	85, 86	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
88	82, 87	oveq12d 7376	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
89	24	addlidd 11334	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
90	88, 89	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
91	80, 90	breqtrd 5124	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))(exp‘𝑥))
92	23, 91	breqdi 5113	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
93		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
94	93, 83	breldm 5857	. . . . . . . . . . 11 ⊢ (𝑥(ℂ D exp)(exp‘𝑥) → 𝑥 ∈ dom (ℂ D exp))
95	92, 94	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ dom (ℂ D exp))
96	95	ssriv 3937	. . . . . . . . 9 ⊢ ℂ ⊆ dom (ℂ D exp)
97	2, 96	eqssi 3950	. . . . . . . 8 ⊢ dom (ℂ D exp) = ℂ
98	97	feq2i 6654	. . . . . . 7 ⊢ ((ℂ D exp):dom (ℂ D exp)⟶ℂ ↔ (ℂ D exp):ℂ⟶ℂ)
99	1, 98	mpbi 230	. . . . . 6 ⊢ (ℂ D exp):ℂ⟶ℂ
100	99	a1i 11	. . . . 5 ⊢ (⊤ → (ℂ D exp):ℂ⟶ℂ)
101	100	feqmptd 6902	. . . 4 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
102		ffun 6665	. . . . . . 7 ⊢ ((ℂ D exp):dom (ℂ D exp)⟶ℂ → Fun (ℂ D exp))
103	1, 102	ax-mp 5	. . . . . 6 ⊢ Fun (ℂ D exp)
104		funbrfv 6882	. . . . . 6 ⊢ (Fun (ℂ D exp) → (𝑥(ℂ D exp)(exp‘𝑥) → ((ℂ D exp)‘𝑥) = (exp‘𝑥)))
105	103, 92, 104	mpsyl 68	. . . . 5 ⊢ (𝑥 ∈ ℂ → ((ℂ D exp)‘𝑥) = (exp‘𝑥))
106	105	mpteq2ia 5193	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
107	101, 106	eqtrdi 2787	. . 3 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
108	19	a1i 11	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
109	108	feqmptd 6902	. . 3 ⊢ (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
110	107, 109	eqtr4d 2774	. 2 ⊢ (⊤ → (ℂ D exp) = exp)
111	110	mptru 1548	1 ⊢ (ℂ D exp) = exp

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2113 Vcvv 3440 {csn 4580 {cpr 4582 ⟨cop 4586 class class class wbr 5098 ↦ cmpt 5179 × cxp 5622 dom cdm 5624 ∘ ccom 5628 Fun wfun 6486 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ∘_f cof 7620 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 − cmin 11364 expce 15984 TopOpenctopn 17341 ℂ_fldccnfld 21309 D cdv 25820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824

This theorem is referenced by: dvsincos 25941 efcn 26409 efcvx 26415 pige3ALT 26485 dvrelog 26602 dvlog 26616 dvcxp1 26705 dvcxp2 26706 dvcncxp1 26708 itgexpif 34763 dvsef 44583 expgrowthi 44584 expgrowth 44586

W3C validator