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Theorem dvcxp1 26237

Description: The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.)

**Assertion**
Ref	Expression
dvcxp1	⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐𝐴))) = (𝑥 ∈ ℝ⁺ ↦ (𝐴 · (𝑥↑_𝑐(𝐴 − 1)))))

Distinct variable group: 𝑥,𝐴

Proof of Theorem dvcxp1 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reelprrecn 11198	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ ℂ → ℝ ∈ {ℝ, ℂ})
3		relogcl 26075	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
4	3	adantl 482	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
5		rpreccl 12996	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
6	5	adantl 482	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
7		recn 11196	. . . 4 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
8		mulcl 11190	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
9		efcl 16022	. . . . 5 ⊢ ((𝐴 · 𝑦) ∈ ℂ → (exp‘(𝐴 · 𝑦)) ∈ ℂ)
10	8, 9	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (exp‘(𝐴 · 𝑦)) ∈ ℂ)
11	7, 10	sylan2 593	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℝ) → (exp‘(𝐴 · 𝑦)) ∈ ℂ)
12		ovexd 7440	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℝ) → ((exp‘(𝐴 · 𝑦)) · 𝐴) ∈ V)
13		relogf1o 26066	. . . . . . . 8 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
14		f1of 6830	. . . . . . . 8 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
15	13, 14	mp1i 13	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
16	15	feqmptd 6957	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
17		fvres 6907	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
18	17	mpteq2ia 5250	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
19	16, 18	eqtrdi 2788	. . . . 5 ⊢ (𝐴 ∈ ℂ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
20	19	oveq2d 7421	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
21		dvrelog 26136	. . . 4 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
22	20, 21	eqtr3di 2787	. . 3 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
23		eqid 2732	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
24	23	cnfldtopon 24290	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
25		toponmax 22419	. . . . 5 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂ_fld))
26	24, 25	mp1i 13	. . . 4 ⊢ (𝐴 ∈ ℂ → ℂ ∈ (TopOpen‘ℂ_fld))
27		ax-resscn 11163	. . . . . 6 ⊢ ℝ ⊆ ℂ
28	27	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → ℝ ⊆ ℂ)
29		df-ss 3964	. . . . 5 ⊢ (ℝ ⊆ ℂ ↔ (ℝ ∩ ℂ) = ℝ)
30	28, 29	sylib 217	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℝ ∩ ℂ) = ℝ)
31		ovexd 7440	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((exp‘(𝐴 · 𝑦)) · 𝐴) ∈ V)
32		cnelprrecn 11199	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
33	32	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → ℂ ∈ {ℝ, ℂ})
34		simpl 483	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 𝐴 ∈ ℂ)
35		efcl 16022	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
36	35	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
37		simpr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
38		1cnd 11205	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
39	33	dvmptid 25465	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
40		id 22	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
41	33, 37, 38, 39, 40	dvmptcmul 25472	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (𝐴 · 1)))
42		mulrid 11208	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
43	42	mpteq2dv 5249	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 1)) = (𝑦 ∈ ℂ ↦ 𝐴))
44	41, 43	eqtrd 2772	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ 𝐴))
45		dvef 25488	. . . . . 6 ⊢ (ℂ D exp) = exp
46		eff 16021	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
47	46	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → exp:ℂ⟶ℂ)
48	47	feqmptd 6957	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
49	48	eqcomd 2738	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (exp‘𝑥)) = exp)
50	49	oveq2d 7421	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘𝑥))) = (ℂ D exp))
51	45, 50, 49	3eqtr4a 2798	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘𝑥))) = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
52		fveq2 6888	. . . . 5 ⊢ (𝑥 = (𝐴 · 𝑦) → (exp‘𝑥) = (exp‘(𝐴 · 𝑦)))
53	33, 33, 8, 34, 36, 36, 44, 51, 52, 52	dvmptco 25480	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ ((exp‘(𝐴 · 𝑦)) · 𝐴)))
54	23, 2, 26, 30, 10, 31, 53	dvmptres3 25464	. . 3 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑦 ∈ ℝ ↦ (exp‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((exp‘(𝐴 · 𝑦)) · 𝐴)))
55		oveq2 7413	. . . 4 ⊢ (𝑦 = (log‘𝑥) → (𝐴 · 𝑦) = (𝐴 · (log‘𝑥)))
56	55	fveq2d 6892	. . 3 ⊢ (𝑦 = (log‘𝑥) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · (log‘𝑥))))
57	56	oveq1d 7420	. . 3 ⊢ (𝑦 = (log‘𝑥) → ((exp‘(𝐴 · 𝑦)) · 𝐴) = ((exp‘(𝐴 · (log‘𝑥))) · 𝐴))
58	2, 2, 4, 6, 11, 12, 22, 54, 56, 57	dvmptco 25480	. 2 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥))))) = (𝑥 ∈ ℝ⁺ ↦ (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥))))
59		rpcn 12980	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
60	59	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
61		rpne0 12986	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
62	61	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
63		simpl 483	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
64	60, 62, 63	cxpefd 26211	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑥))))
65	64	mpteq2dva 5247	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐𝐴)) = (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥)))))
66	65	oveq2d 7421	. 2 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐𝐴))) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥))))))
67		1cnd 11205	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
68	60, 62, 63, 67	cxpsubd 26217	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐴 − 1)) = ((𝑥↑_𝑐𝐴) / (𝑥↑_𝑐1)))
69	60	cxp1d 26205	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐1) = 𝑥)
70	69	oveq2d 7421	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐𝐴) / (𝑥↑_𝑐1)) = ((𝑥↑_𝑐𝐴) / 𝑥))
71	60, 63	cxpcld 26207	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐𝐴) ∈ ℂ)
72	71, 60, 62	divrecd 11989	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐𝐴) / 𝑥) = ((𝑥↑_𝑐𝐴) · (1 / 𝑥)))
73	68, 70, 72	3eqtrd 2776	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐴 − 1)) = ((𝑥↑_𝑐𝐴) · (1 / 𝑥)))
74	73	oveq2d 7421	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · (𝑥↑_𝑐(𝐴 − 1))) = (𝐴 · ((𝑥↑_𝑐𝐴) · (1 / 𝑥))))
75	6	rpcnd 13014	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
76	63, 71, 75	mul12d 11419	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · ((𝑥↑_𝑐𝐴) · (1 / 𝑥))) = ((𝑥↑_𝑐𝐴) · (𝐴 · (1 / 𝑥))))
77	71, 63, 75	mulassd 11233	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (((𝑥↑_𝑐𝐴) · 𝐴) · (1 / 𝑥)) = ((𝑥↑_𝑐𝐴) · (𝐴 · (1 / 𝑥))))
78	76, 77	eqtr4d 2775	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · ((𝑥↑_𝑐𝐴) · (1 / 𝑥))) = (((𝑥↑_𝑐𝐴) · 𝐴) · (1 / 𝑥)))
79	64	oveq1d 7420	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐𝐴) · 𝐴) = ((exp‘(𝐴 · (log‘𝑥))) · 𝐴))
80	79	oveq1d 7420	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (((𝑥↑_𝑐𝐴) · 𝐴) · (1 / 𝑥)) = (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥)))
81	74, 78, 80	3eqtrd 2776	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · (𝑥↑_𝑐(𝐴 − 1))) = (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥)))
82	81	mpteq2dva 5247	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (𝐴 · (𝑥↑_𝑐(𝐴 − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥))))
83	58, 66, 82	3eqtr4d 2782	1 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐𝐴))) = (𝑥 ∈ ℝ⁺ ↦ (𝐴 · (𝑥↑_𝑐(𝐴 − 1)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 {cpr 4629 ↦ cmpt 5230 ↾ cres 5677 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 − cmin 11440 / cdiv 11867 ℝ⁺crp 12970 expce 16001 TopOpenctopn 17363 ℂ_fldccnfld 20936 TopOnctopon 22403 D cdv 25371 logclog 26054 ↑_𝑐ccxp 26055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-cxp 26057

This theorem is referenced by: dvsqrt 26239 logdivsqrle 33650

W3C validator