dvcxp1 - Metamath Proof Explorer

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Theorem dvcxp1 26668

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 11225	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ ℂ → ℝ ∈ {ℝ, ℂ})
3		relogcl 26503	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
4	3	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
5		rpreccl 13027	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
6	5	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
7		recn 11223	. . . 4 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
8		mulcl 11217	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
9		efcl 16053	. . . . 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 592	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℝ) → (exp‘(𝐴 · 𝑦)) ∈ ℂ)
12		ovexd 7450	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℝ) → ((exp‘(𝐴 · 𝑦)) · 𝐴) ∈ V)
13		relogf1o 26494	. . . . . . . 8 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
14		f1of 6834	. . . . . . . 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 6962	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
17		fvres 6911	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
18	17	mpteq2ia 5246	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
19	16, 18	eqtrdi 2784	. . . . 5 ⊢ (𝐴 ∈ ℂ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
20	19	oveq2d 7431	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
21		dvrelog 26565	. . . 4 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
22	20, 21	eqtr3di 2783	. . 3 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
23		eqid 2728	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
24	23	cnfldtopon 24693	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
25		toponmax 22822	. . . . 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 11190	. . . . . 6 ⊢ ℝ ⊆ ℂ
28	27	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → ℝ ⊆ ℂ)
29		df-ss 3962	. . . . 5 ⊢ (ℝ ⊆ ℂ ↔ (ℝ ∩ ℂ) = ℝ)
30	28, 29	sylib 217	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℝ ∩ ℂ) = ℝ)
31		ovexd 7450	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((exp‘(𝐴 · 𝑦)) · 𝐴) ∈ V)
32		cnelprrecn 11226	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
33	32	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → ℂ ∈ {ℝ, ℂ})
34		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 𝐴 ∈ ℂ)
35		efcl 16053	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
36	35	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (exp‘𝑥) ∈ ℂ)
37		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
38		1cnd 11234	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
39	33	dvmptid 25883	. . . . . . 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 25890	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (𝐴 · 1)))
42		mulrid 11237	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
43	42	mpteq2dv 5245	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 1)) = (𝑦 ∈ ℂ ↦ 𝐴))
44	41, 43	eqtrd 2768	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ 𝐴))
45		dvef 25906	. . . . . 6 ⊢ (ℂ D exp) = exp
46		eff 16052	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
47	46	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → exp:ℂ⟶ℂ)
48	47	feqmptd 6962	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
49	48	eqcomd 2734	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (exp‘𝑥)) = exp)
50	49	oveq2d 7431	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘𝑥))) = (ℂ D exp))
51	45, 50, 49	3eqtr4a 2794	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘𝑥))) = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
52		fveq2 6892	. . . . 5 ⊢ (𝑥 = (𝐴 · 𝑦) → (exp‘𝑥) = (exp‘(𝐴 · 𝑦)))
53	33, 33, 8, 34, 36, 36, 44, 51, 52, 52	dvmptco 25898	. . . 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 25882	. . 3 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑦 ∈ ℝ ↦ (exp‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((exp‘(𝐴 · 𝑦)) · 𝐴)))
55		oveq2 7423	. . . 4 ⊢ (𝑦 = (log‘𝑥) → (𝐴 · 𝑦) = (𝐴 · (log‘𝑥)))
56	55	fveq2d 6896	. . 3 ⊢ (𝑦 = (log‘𝑥) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · (log‘𝑥))))
57	56	oveq1d 7430	. . 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 25898	. 2 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥))))) = (𝑥 ∈ ℝ⁺ ↦ (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥))))
59		rpcn 13011	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
60	59	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
61		rpne0 13017	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
62	61	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
63		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
64	60, 62, 63	cxpefd 26640	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑥))))
65	64	mpteq2dva 5243	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐𝐴)) = (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥)))))
66	65	oveq2d 7431	. 2 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐𝐴))) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥))))))
67		1cnd 11234	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
68	60, 62, 63, 67	cxpsubd 26646	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐴 − 1)) = ((𝑥↑_𝑐𝐴) / (𝑥↑_𝑐1)))
69	60	cxp1d 26634	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐1) = 𝑥)
70	69	oveq2d 7431	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐𝐴) / (𝑥↑_𝑐1)) = ((𝑥↑_𝑐𝐴) / 𝑥))
71	60, 63	cxpcld 26636	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐𝐴) ∈ ℂ)
72	71, 60, 62	divrecd 12018	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐𝐴) / 𝑥) = ((𝑥↑_𝑐𝐴) · (1 / 𝑥)))
73	68, 70, 72	3eqtrd 2772	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐴 − 1)) = ((𝑥↑_𝑐𝐴) · (1 / 𝑥)))
74	73	oveq2d 7431	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · (𝑥↑_𝑐(𝐴 − 1))) = (𝐴 · ((𝑥↑_𝑐𝐴) · (1 / 𝑥))))
75	6	rpcnd 13045	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
76	63, 71, 75	mul12d 11448	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · ((𝑥↑_𝑐𝐴) · (1 / 𝑥))) = ((𝑥↑_𝑐𝐴) · (𝐴 · (1 / 𝑥))))
77	71, 63, 75	mulassd 11262	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (((𝑥↑_𝑐𝐴) · 𝐴) · (1 / 𝑥)) = ((𝑥↑_𝑐𝐴) · (𝐴 · (1 / 𝑥))))
78	76, 77	eqtr4d 2771	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · ((𝑥↑_𝑐𝐴) · (1 / 𝑥))) = (((𝑥↑_𝑐𝐴) · 𝐴) · (1 / 𝑥)))
79	64	oveq1d 7430	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐𝐴) · 𝐴) = ((exp‘(𝐴 · (log‘𝑥))) · 𝐴))
80	79	oveq1d 7430	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (((𝑥↑_𝑐𝐴) · 𝐴) · (1 / 𝑥)) = (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥)))
81	74, 78, 80	3eqtrd 2772	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · (𝑥↑_𝑐(𝐴 − 1))) = (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥)))
82	81	mpteq2dva 5243	. 2 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (𝐴 · (𝑥↑_𝑐(𝐴 − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥))))
83	58, 66, 82	3eqtr4d 2778	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 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 Vcvv 3470 ∩ cin 3944 ⊆ wss 3945 {cpr 4627 ↦ cmpt 5226 ↾ cres 5675 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7415 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 · cmul 11138 − cmin 11469 / cdiv 11896 ℝ⁺crp 13001 expce 16032 TopOpenctopn 17397 ℂ_fldccnfld 21273 TopOnctopon 22806 D cdv 25786 logclog 26482 ↑_𝑐ccxp 26483

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-supp 8161 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-er 8719 df-map 8841 df-pm 8842 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-fac 14260 df-bc 14289 df-hash 14317 df-shft 15041 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660 df-ef 16038 df-sin 16040 df-cos 16041 df-pi 16043 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19018 df-cntz 19262 df-cmn 19731 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-fbas 21270 df-fg 21271 df-cnfld 21274 df-top 22790 df-topon 22807 df-topsp 22829 df-bases 22843 df-cld 22917 df-ntr 22918 df-cls 22919 df-nei 22996 df-lp 23034 df-perf 23035 df-cn 23125 df-cnp 23126 df-haus 23213 df-cmp 23285 df-tx 23460 df-hmeo 23653 df-fil 23744 df-fm 23836 df-flim 23837 df-flf 23838 df-xms 24220 df-ms 24221 df-tms 24222 df-cncf 24792 df-limc 25789 df-dv 25790 df-log 26484 df-cxp 26485

This theorem is referenced by: dvsqrt 26670 logdivsqrle 34277

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