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Theorem dvcxp1 26472

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 11204	. . . 4 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . 3 ⊢ (𝐴 ∈ ℂ → ℝ ∈ {ℝ, ℂ})
3		relogcl 26308	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
4	3	adantl 482	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
5		rpreccl 13004	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
6	5	adantl 482	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
7		recn 11202	. . . 4 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
8		mulcl 11196	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
9		efcl 16030	. . . . 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 7446	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℝ) → ((exp‘(𝐴 · 𝑦)) · 𝐴) ∈ V)
13		relogf1o 26299	. . . . . . . 8 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
14		f1of 6833	. . . . . . . 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 6960	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
17		fvres 6910	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
18	17	mpteq2ia 5251	. . . . . 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 7427	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
21		dvrelog 26369	. . . 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 24519	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
25		toponmax 22648	. . . . 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 11169	. . . . . 6 ⊢ ℝ ⊆ ℂ
28	27	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → ℝ ⊆ ℂ)
29		df-ss 3965	. . . . 5 ⊢ (ℝ ⊆ ℂ ↔ (ℝ ∩ ℂ) = ℝ)
30	28, 29	sylib 217	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℝ ∩ ℂ) = ℝ)
31		ovexd 7446	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((exp‘(𝐴 · 𝑦)) · 𝐴) ∈ V)
32		cnelprrecn 11205	. . . . . 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 16030	. . . . . 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 11213	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
39	33	dvmptid 25698	. . . . . . 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 25705	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ (𝐴 · 1)))
42		mulrid 11216	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴)
43	42	mpteq2dv 5250	. . . . . 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 25721	. . . . . 6 ⊢ (ℂ D exp) = exp
46		eff 16029	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
47	46	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → exp:ℂ⟶ℂ)
48	47	feqmptd 6960	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → exp = (𝑥 ∈ ℂ ↦ (exp‘𝑥)))
49	48	eqcomd 2738	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (exp‘𝑥)) = exp)
50	49	oveq2d 7427	. . . . . 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 6891	. . . . 5 ⊢ (𝑥 = (𝐴 · 𝑦) → (exp‘𝑥) = (exp‘(𝐴 · 𝑦)))
53	33, 33, 8, 34, 36, 36, 44, 51, 52, 52	dvmptco 25713	. . . 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 25697	. . 3 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑦 ∈ ℝ ↦ (exp‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((exp‘(𝐴 · 𝑦)) · 𝐴)))
55		oveq2 7419	. . . 4 ⊢ (𝑦 = (log‘𝑥) → (𝐴 · 𝑦) = (𝐴 · (log‘𝑥)))
56	55	fveq2d 6895	. . 3 ⊢ (𝑦 = (log‘𝑥) → (exp‘(𝐴 · 𝑦)) = (exp‘(𝐴 · (log‘𝑥))))
57	56	oveq1d 7426	. . 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 25713	. 2 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥))))) = (𝑥 ∈ ℝ⁺ ↦ (((exp‘(𝐴 · (log‘𝑥))) · 𝐴) · (1 / 𝑥))))
59		rpcn 12988	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
60	59	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
61		rpne0 12994	. . . . . 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 26444	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑥))))
65	64	mpteq2dva 5248	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐𝐴)) = (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥)))))
66	65	oveq2d 7427	. 2 ⊢ (𝐴 ∈ ℂ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥↑_𝑐𝐴))) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (exp‘(𝐴 · (log‘𝑥))))))
67		1cnd 11213	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
68	60, 62, 63, 67	cxpsubd 26450	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐴 − 1)) = ((𝑥↑_𝑐𝐴) / (𝑥↑_𝑐1)))
69	60	cxp1d 26438	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐1) = 𝑥)
70	69	oveq2d 7427	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐𝐴) / (𝑥↑_𝑐1)) = ((𝑥↑_𝑐𝐴) / 𝑥))
71	60, 63	cxpcld 26440	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐𝐴) ∈ ℂ)
72	71, 60, 62	divrecd 11997	. . . . . 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 7427	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · (𝑥↑_𝑐(𝐴 − 1))) = (𝐴 · ((𝑥↑_𝑐𝐴) · (1 / 𝑥))))
75	6	rpcnd 13022	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
76	63, 71, 75	mul12d 11427	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → (𝐴 · ((𝑥↑_𝑐𝐴) · (1 / 𝑥))) = ((𝑥↑_𝑐𝐴) · (𝐴 · (1 / 𝑥))))
77	71, 63, 75	mulassd 11241	. . . . 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 7426	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑_𝑐𝐴) · 𝐴) = ((exp‘(𝐴 · (log‘𝑥))) · 𝐴))
80	79	oveq1d 7426	. . . 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 5248	. 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 3947 ⊆ wss 3948 {cpr 4630 ↦ cmpt 5231 ↾ cres 5678 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 − cmin 11448 / cdiv 11875 ℝ⁺crp 12978 expce 16009 TopOpenctopn 17371 ℂ_fldccnfld 21144 TopOnctopon 22632 D cdv 25604 logclog 26287 ↑_𝑐ccxp 26288

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

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 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 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ioc 13333 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-fac 14238 df-bc 14267 df-hash 14295 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-fbas 21141 df-fg 21142 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 df-perf 22861 df-cn 22951 df-cnp 22952 df-haus 23039 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-fil 23570 df-fm 23662 df-flim 23663 df-flf 23664 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-limc 25607 df-dv 25608 df-log 26289 df-cxp 26290

This theorem is referenced by: dvsqrt 26474 logdivsqrle 33948

W3C validator