dvrelog2b - Mathbox for metakunt

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Theorem dvrelog2b 39666

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 11765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2ne0 11791	. . . . . . . . 9 ⊢ 2 ≠ 0
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
6		1red 10693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
7		1lt2 11858	. . . . . . . . . . 11 ⊢ 1 < 2
8	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
9	6, 8	ltned 10827	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
10	9	necomd 3006	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
11	5, 10	nelprd 4556	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
12	3, 11	eldifd 3871	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
13		elioore 12822	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
14		recn 10678	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℂ)
16	15	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
17		elsni 4542	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
18		dvrelog2b.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 ≤ 𝐴)
19		0xr 10739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℝ^*
20	19	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℝ^*)
21		dvrelog2b.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		xrlenlt 10757	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
23	20, 21, 22	syl2anc 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
24	18, 23	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝐴 < 0)
25	24	orcd 870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵))
26		ianor 979	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐴 < 0 ∧ 0 < 𝐵) ↔ (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵))
27	25, 26	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ (𝐴 < 0 ∧ 0 < 𝐵))
28		dvrelog2b.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
29		elioo5 12849	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵)))
30	21, 28, 20, 29	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵)))
31	30	notbid 321	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (¬ 0 ∈ (𝐴(,)𝐵) ↔ ¬ (𝐴 < 0 ∧ 0 < 𝐵)))
32	27, 31	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
33	32	a1d 25	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 ∈ (𝐴(,)𝐵) → ¬ 0 ∈ (𝐴(,)𝐵)))
34	33	imp 410	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ (𝐴(,)𝐵))
35	34	pm2.01da 798	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
36	35	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
37		eleq1 2839	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵)))
38	37	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 0) → (𝑥 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵)))
39	36, 38	mtbird 328	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 𝑥 ∈ (𝐴(,)𝐵))
40	17, 39	sylan2 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {0}) → ¬ 𝑥 ∈ (𝐴(,)𝐵))
41	40	ex 416	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ {0} → ¬ 𝑥 ∈ (𝐴(,)𝐵)))
42	41	con2d 136	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 ∈ {0}))
43	42	imp 410	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0})
44	16, 43	eldifd 3871	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
45		logbval 25464	. . . . . 6 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
46	12, 44, 45	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
47	46	mpteq2dva 5131	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
48	2, 47	eqtrd 2793	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
49	48	oveq2d 7172	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))))
50		reelprrecn 10680	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
51	50	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
52	39	ex 416	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 0 → ¬ 𝑥 ∈ (𝐴(,)𝐵)))
53	52	con2d 136	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 0))
54		biidd 265	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝑥 = 0 ↔ 𝑥 = 0))
55	54	necon3bbid 2988	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (¬ 𝑥 = 0 ↔ 𝑥 ≠ 0))
56	55	pm5.74i 274	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 0) ↔ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0))
57	53, 56	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0))
58	57	imp 410	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
59	16, 58	logcld 25274	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
60	13	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
61	6, 60, 58	redivcld 11519	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
62		dvrelog2b.4	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
63		eqid 2758	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))
64		eqid 2758	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))
65	21, 28, 18, 62, 63, 64	dvrelog3 39665	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
66		2cnd 11765	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
67	4	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
68	66, 67	logcld 25274	. . . 4 ⊢ (𝜑 → (log‘2) ∈ ℂ)
69		0red 10695	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
70		2rp 12448	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
71		loggt0b 25335	. . . . . . . . 9 ⊢ (2 ∈ ℝ⁺ → (0 < (log‘2) ↔ 1 < 2))
72	70, 71	ax-mp 5	. . . . . . . 8 ⊢ (0 < (log‘2) ↔ 1 < 2)
73	7, 72	mpbir 234	. . . . . . 7 ⊢ 0 < (log‘2)
74	73	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 < (log‘2))
75	69, 74	ltned 10827	. . . . 5 ⊢ (𝜑 → 0 ≠ (log‘2))
76	75	necomd 3006	. . . 4 ⊢ (𝜑 → (log‘2) ≠ 0)
77	51, 59, 61, 65, 68, 76	dvmptdivc 24677	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))))
78	3, 5	logcld 25274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
79	76	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
80	16, 78, 58, 79	recdiv2d 11485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((1 / 𝑥) / (log‘2)) = (1 / (𝑥 · (log‘2))))
81	80	mpteq2dva 5131	. . . 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 2764	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = 𝐺)
85	81, 84	eqtrd 2793	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))) = 𝐺)
86	77, 85	eqtrd 2793	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = 𝐺)
87	49, 86	eqtrd 2793	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∖ cdif 3857 {csn 4525 {cpr 4527 class class class wbr 5036 ↦ cmpt 5116 ‘cfv 6340 (class class class)co 7156 ℂcc 10586 ℝcr 10587 0cc0 10588 1c1 10589 · cmul 10593 ℝ^*cxr 10725 < clt 10726 ≤ cle 10727 / cdiv 11348 2c2 11742 ℝ⁺crp 12443 (,)cioo 12792 D cdv 24575 logclog 25258 log_b clogb 25462

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-inf2 9150 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666 ax-addf 10667 ax-mulf 10668

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7411 df-om 7586 df-1st 7699 df-2nd 7700 df-supp 7842 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-2o 8119 df-er 8305 df-map 8424 df-pm 8425 df-ixp 8493 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fsupp 8880 df-fi 8921 df-sup 8952 df-inf 8953 df-oi 9020 df-card 9414 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-q 12402 df-rp 12444 df-xneg 12561 df-xadd 12562 df-xmul 12563 df-ioo 12796 df-ioc 12797 df-ico 12798 df-icc 12799 df-fz 12953 df-fzo 13096 df-fl 13224 df-mod 13300 df-seq 13432 df-exp 13493 df-fac 13697 df-bc 13726 df-hash 13754 df-shft 14487 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-limsup 14889 df-clim 14906 df-rlim 14907 df-sum 15104 df-ef 15482 df-sin 15484 df-cos 15485 df-pi 15487 df-struct 16556 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-ress 16562 df-plusg 16649 df-mulr 16650 df-starv 16651 df-sca 16652 df-vsca 16653 df-ip 16654 df-tset 16655 df-ple 16656 df-ds 16658 df-unif 16659 df-hom 16660 df-cco 16661 df-rest 16767 df-topn 16768 df-0g 16786 df-gsum 16787 df-topgen 16788 df-pt 16789 df-prds 16792 df-xrs 16846 df-qtop 16851 df-imas 16852 df-xps 16854 df-mre 16928 df-mrc 16929 df-acs 16931 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-submnd 18036 df-mulg 18305 df-cntz 18527 df-cmn 18988 df-psmet 20171 df-xmet 20172 df-met 20173 df-bl 20174 df-mopn 20175 df-fbas 20176 df-fg 20177 df-cnfld 20180 df-top 21607 df-topon 21624 df-topsp 21646 df-bases 21659 df-cld 21732 df-ntr 21733 df-cls 21734 df-nei 21811 df-lp 21849 df-perf 21850 df-cn 21940 df-cnp 21941 df-haus 22028 df-cmp 22100 df-tx 22275 df-hmeo 22468 df-fil 22559 df-fm 22651 df-flim 22652 df-flf 22653 df-xms 23035 df-ms 23036 df-tms 23037 df-cncf 23592 df-limc 24578 df-dv 24579 df-log 25260 df-logb 25463

This theorem is referenced by: dvrelogpow2b 39668 aks4d1p1p6 39673

W3C validator