dvrelog2b - Mathbox for metakunt

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Theorem dvrelog2b 40002

Description: Derivative of the binary logarithm. (Contributed by metakunt, 11-Aug-2024.)

**Hypotheses**
Ref	Expression
dvrelog2b.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
dvrelog2b.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
dvrelog2b.3	⊢ (𝜑 → 0 ≤ 𝐴)
dvrelog2b.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)
dvrelog2b.5	⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
dvrelog2b.6	⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))

**Assertion**
Ref	Expression
dvrelog2b	⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥 Allowed substitution hints: 𝐹(𝑥) 𝐺(𝑥)

**Proof of Theorem dvrelog2b**
Step	Hyp	Ref	Expression
1		dvrelog2b.5	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)))
3		2cnd 11981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2ne0 12007	. . . . . . . . 9 ⊢ 2 ≠ 0
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
6		1red 10907	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
7		1lt2 12074	. . . . . . . . . . 11 ⊢ 1 < 2
8	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
9	6, 8	ltned 11041	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
10	9	necomd 2998	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
11	5, 10	nelprd 4589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
12	3, 11	eldifd 3894	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
13		elioore 13038	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
14		recn 10892	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℂ)
16	15	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
17		elsni 4575	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
18		dvrelog2b.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 ≤ 𝐴)
19		0xr 10953	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℝ^*
20	19	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℝ^*)
21		dvrelog2b.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		xrlenlt 10971	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
23	20, 21, 22	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
24	18, 23	mpbid 231	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝐴 < 0)
25	24	orcd 869	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵))
26		ianor 978	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐴 < 0 ∧ 0 < 𝐵) ↔ (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵))
27	25, 26	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ (𝐴 < 0 ∧ 0 < 𝐵))
28		dvrelog2b.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
29		elioo5 13065	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵)))
30	21, 28, 20, 29	syl3anc 1369	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵)))
31	30	notbid 317	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (¬ 0 ∈ (𝐴(,)𝐵) ↔ ¬ (𝐴 < 0 ∧ 0 < 𝐵)))
32	27, 31	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
33	32	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 ∈ (𝐴(,)𝐵) → ¬ 0 ∈ (𝐴(,)𝐵)))
34	33	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ (𝐴(,)𝐵))
35	34	pm2.01da 795	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
36	35	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
37		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵)))
38	37	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑥 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵)))
39	36, 38	mtbird 324	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 𝑥 ∈ (𝐴(,)𝐵))
40	17, 39	sylan2 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {0}) → ¬ 𝑥 ∈ (𝐴(,)𝐵))
41	40	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ {0} → ¬ 𝑥 ∈ (𝐴(,)𝐵)))
42	41	con2d 134	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 ∈ {0}))
43	42	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0})
44	16, 43	eldifd 3894	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
45		logbval 25821	. . . . . 6 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
46	12, 44, 45	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
47	46	mpteq2dva 5170	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
48	2, 47	eqtrd 2778	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
49	48	oveq2d 7271	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))))
50		reelprrecn 10894	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
51	50	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
52	39	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 0 → ¬ 𝑥 ∈ (𝐴(,)𝐵)))
53	52	con2d 134	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 0))
54		biidd 261	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝑥 = 0 ↔ 𝑥 = 0))
55	54	necon3bbid 2980	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (¬ 𝑥 = 0 ↔ 𝑥 ≠ 0))
56	55	pm5.74i 270	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 0) ↔ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0))
57	53, 56	sylib 217	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0))
58	57	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
59	16, 58	logcld 25631	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
60	13	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
61	6, 60, 58	redivcld 11733	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
62		dvrelog2b.4	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
63		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))
64		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))
65	21, 28, 18, 62, 63, 64	dvrelog3 40001	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
66		2cnd 11981	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
67	4	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
68	66, 67	logcld 25631	. . . 4 ⊢ (𝜑 → (log‘2) ∈ ℂ)
69		0red 10909	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
70		2rp 12664	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
71		loggt0b 25692	. . . . . . . . 9 ⊢ (2 ∈ ℝ⁺ → (0 < (log‘2) ↔ 1 < 2))
72	70, 71	ax-mp 5	. . . . . . . 8 ⊢ (0 < (log‘2) ↔ 1 < 2)
73	7, 72	mpbir 230	. . . . . . 7 ⊢ 0 < (log‘2)
74	73	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 < (log‘2))
75	69, 74	ltned 11041	. . . . 5 ⊢ (𝜑 → 0 ≠ (log‘2))
76	75	necomd 2998	. . . 4 ⊢ (𝜑 → (log‘2) ≠ 0)
77	51, 59, 61, 65, 68, 76	dvmptdivc 25034	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))))
78	3, 5	logcld 25631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
79	76	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
80	16, 78, 58, 79	recdiv2d 11699	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((1 / 𝑥) / (log‘2)) = (1 / (𝑥 · (log‘2))))
81	80	mpteq2dva 5170	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
82		dvrelog2b.6	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))
83	82	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))))
84	83	eqcomd 2744	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = 𝐺)
85	81, 84	eqtrd 2778	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))) = 𝐺)
86	77, 85	eqtrd 2778	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = 𝐺)
87	49, 86	eqtrd 2778	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 {csn 4558 {cpr 4560 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 · cmul 10807 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 / cdiv 11562 2c2 11958 ℝ⁺crp 12659 (,)cioo 13008 D cdv 24932 logclog 25615 log_b clogb 25819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-logb 25820

This theorem is referenced by: dvrelogpow2b 40004 aks4d1p1p6 40009

W3C validator