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Theorem dvef 26033

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 25958	. . . . . . 7 ⊢ (ℂ D exp):dom (ℂ D exp)⟶ℂ
2		dvbsss 25952	. . . . . . . . 9 ⊢ dom (ℂ D exp) ⊆ ℂ
3		subcl 11505	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
4	3	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑧 − 𝑥) ∈ ℂ)
5		efadd 16127	. . . . . . . . . . . . . . . . 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 11514	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 + (𝑧 − 𝑥)) = 𝑧)
8	7	fveq2d 6911	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑥 + (𝑧 − 𝑥))) = (exp‘𝑧))
9	6, 8	eqtr3d 2777	. . . . . . . . . . . . . . 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 11234	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
12	11	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ℂ ∈ V)
13		fvexd 6922	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘𝑥) ∈ V)
14		fvexd 6922	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (exp‘(𝑧 − 𝑥)) ∈ V)
15		fconstmpt 5751	. . . . . . . . . . . . . . . 16 ⊢ (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥))
16	15	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}) = (𝑧 ∈ ℂ ↦ (exp‘𝑥)))
17		eqidd 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
18	12, 13, 14, 16, 17	offval2 7717	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = (𝑧 ∈ ℂ ↦ ((exp‘𝑥) · (exp‘(𝑧 − 𝑥)))))
19		eff 16114	. . . . . . . . . . . . . . . 16 ⊢ exp:ℂ⟶ℂ
20	19	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → exp:ℂ⟶ℂ)
21	20	feqmptd 6977	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → exp = (𝑧 ∈ ℂ ↦ (exp‘𝑧)))
22	10, 18, 21	3eqtr4d 2785	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))) = exp)
23	22	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (ℂ D ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))) = (ℂ D exp))
24		efcl 16115	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
25		fconstg 6796	. . . . . . . . . . . . . . . 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 4814	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → {(exp‘𝑥)} ⊆ ℂ)
28	26, 27	fssd 6754	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (ℂ × {(exp‘𝑥)}):ℂ⟶ℂ)
29		ssidd 4019	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ℂ ⊆ ℂ)
30		efcl 16115	. . . . . . . . . . . . . . . 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 7135	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))):ℂ⟶ℂ)
33		c0ex 11253	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
34	33	snid 4667	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ {0}
35		opelxpi 5726	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℂ ∧ 0 ∈ {0}) → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
36	34, 35	mpan2 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ × {0}))
37		dvconst 25967	. . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 0⟩ ∈ (ℂ D (ℂ × {(exp‘𝑥)})))
40		df-br 5149	. . . . . . . . . . . . . . 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 7152	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))
43	42	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)))) = (ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))
44	4	fmpttd 7135	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)):ℂ⟶ℂ)
45		oveq1 7438	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑥 → (𝑧 − 𝑥) = (𝑥 − 𝑥))
46		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥)) = (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))
47		ovex 7464	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 − 𝑥) ∈ V
48	45, 46, 47	fvmpt 7016	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = (𝑥 − 𝑥))
49		subid 11526	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (𝑥 − 𝑥) = 0)
50	48, 49	eqtrd 2775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥) = 0)
51		dveflem 26032	. . . . . . . . . . . . . . . . . 18 ⊢ 0(ℂ D exp)1
52	50, 51	eqbrtrdi 5187	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))‘𝑥)(ℂ D exp)1)
53		1ex 11255	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ V
54	53	snid 4667	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ {1}
55		opelxpi 5726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ {1}) → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
56	54, 55	mpan2 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ × {1}))
57		cnelprrecn 11246	. . . . . . . . . . . . . . . . . . . . . 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 11254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 1 ∈ ℂ)
61	58	dvmptid 26010	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑧)) = (𝑧 ∈ ℂ ↦ 1))
62		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 𝑥 ∈ ℂ)
63		0cnd 11252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → 0 ∈ ℂ)
64		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
65	58, 64	dvmptc 26011	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ 𝑥)) = (𝑧 ∈ ℂ ↦ 0))
66	58, 59, 60, 61, 62, 63, 65	dvmptsub 26020	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (𝑧 ∈ ℂ ↦ (1 − 0)))
67		1m0e1 12385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 − 0) = 1
68	67	mpteq2i 5253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (𝑧 ∈ ℂ ↦ 1)
69		fconstmpt 5751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℂ × {1}) = (𝑧 ∈ ℂ ↦ 1)
70	68, 69	eqtr4i 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℂ ↦ (1 − 0)) = (ℂ × {1})
71	66, 70	eqtrdi 2791	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))) = (ℂ × {1}))
72	56, 71	eleqtrrd 2842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ⟨𝑥, 1⟩ ∈ (ℂ D (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))
73		df-br 5149	. . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
76	20, 29, 44, 29, 29, 29, 52, 74, 75	dvcobr 25998	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))(1 · 1))
77		1t1e1 12426	. . . . . . . . . . . . . . . 16 ⊢ (1 · 1) = 1
78	76, 77	breqtrdi 5189	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (exp ∘ (𝑧 ∈ ℂ ↦ (𝑧 − 𝑥))))1)
79	43, 78	breqdi 5163	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥))))1)
80	28, 29, 32, 29, 29, 41, 79, 75	dvmulbr 25990	. . . . . . . . . . . . 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 7105	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥) ∈ ℂ)
82	81	mul02d 11457	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) = 0)
83		fvex 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (exp‘𝑥) ∈ V
84	83	fvconst2 7224	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((ℂ × {(exp‘𝑥)})‘𝑥) = (exp‘𝑥))
85	84	oveq2d 7447	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (1 · (exp‘𝑥)))
86	24	mullidd 11277	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 · (exp‘𝑥)) = (exp‘𝑥))
87	85, 86	eqtrd 2775	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → (1 · ((ℂ × {(exp‘𝑥)})‘𝑥)) = (exp‘𝑥))
88	82, 87	oveq12d 7449	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (0 + (exp‘𝑥)))
89	24	addlidd 11460	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (0 + (exp‘𝑥)) = (exp‘𝑥))
90	88, 89	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → ((0 · ((𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))‘𝑥)) + (1 · ((ℂ × {(exp‘𝑥)})‘𝑥))) = (exp‘𝑥))
91	80, 90	breqtrd 5174	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D ((ℂ × {(exp‘𝑥)}) ∘_f · (𝑧 ∈ ℂ ↦ (exp‘(𝑧 − 𝑥)))))(exp‘𝑥))
92	23, 91	breqdi 5163	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → 𝑥(ℂ D exp)(exp‘𝑥))
93		vex 3482	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
94	93, 83	breldm 5922	. . . . . . . . . . 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 3999	. . . . . . . . 9 ⊢ ℂ ⊆ dom (ℂ D exp)
97	2, 96	eqssi 4012	. . . . . . . 8 ⊢ dom (ℂ D exp) = ℂ
98	97	feq2i 6729	. . . . . . 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 6977	. . . 4 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)))
102		ffun 6740	. . . . . . 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 6958	. . . . . 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 5251	. . . 4 ⊢ (𝑥 ∈ ℂ ↦ ((ℂ D exp)‘𝑥)) = (𝑥 ∈ ℂ ↦ (exp‘𝑥))
107	101, 106	eqtrdi 2791	. . 3 ⊢ (⊤ → (ℂ D exp) = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
108	19	a1i 11	. . . 4 ⊢ (⊤ → exp:ℂ⟶ℂ)
109	108	feqmptd 6977	. . 3 ⊢ (⊤ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
110	107, 109	eqtr4d 2778	. 2 ⊢ (⊤ → (ℂ D exp) = exp)
111	110	mptru 1544	1 ⊢ (ℂ D exp) = exp

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 Vcvv 3478 {csn 4631 {cpr 4633 ⟨cop 4637 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 dom cdm 5689 ∘ ccom 5693 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘_f cof 7695 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 expce 16094 TopOpenctopn 17468 ℂ_fldccnfld 21382 D cdv 25913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917

This theorem is referenced by: dvsincos 26034 efcn 26502 efcvx 26508 pige3ALT 26577 dvrelog 26694 dvlog 26708 dvcxp1 26797 dvcxp2 26798 dvcncxp1 26800 itgexpif 34600 dvsef 44328 expgrowthi 44329 expgrowth 44331

W3C validator