dvrelog2b - Mathbox for metakunt

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Theorem dvrelog2b 42320

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 12223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2ne0 12249	. . . . . . . . 9 ⊢ 2 ≠ 0
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
6		1red 11133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
7		1lt2 12311	. . . . . . . . . . 11 ⊢ 1 < 2
8	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
9	6, 8	ltned 11269	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
10	9	necomd 2987	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
11	5, 10	nelprd 4614	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
12	3, 11	eldifd 3912	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
13		elioore 13291	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
14		recn 11116	. . . . . . . . 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 4597	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
18		dvrelog2b.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 ≤ 𝐴)
19		0xr 11179	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℝ^*
20	19	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℝ^*)
21		dvrelog2b.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		xrlenlt 11197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
23	20, 21, 22	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
24	18, 23	mpbid 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝐴 < 0)
25	24	orcd 873	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵))
26		ianor 983	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (𝐴 < 0 ∧ 0 < 𝐵) ↔ (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵))
27	25, 26	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ (𝐴 < 0 ∧ 0 < 𝐵))
28		dvrelog2b.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
29		elioo5 13319	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵)))
30	21, 28, 20, 29	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵)))
31	30	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (¬ 0 ∈ (𝐴(,)𝐵) ↔ ¬ (𝐴 < 0 ∧ 0 < 𝐵)))
32	27, 31	mpbird 257	. . . . . . . . . . . . . . . 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 798	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
36	35	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
37		eleq1 2824	. . . . . . . . . . . . 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 325	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 𝑥 ∈ (𝐴(,)𝐵))
40	17, 39	sylan2 593	. . . . . . . . . 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 3912	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
45		logbval 26732	. . . . . 6 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
46	12, 44, 45	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
47	46	mpteq2dva 5191	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
48	2, 47	eqtrd 2771	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
49	48	oveq2d 7374	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))))
50		reelprrecn 11118	. . . . 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 262	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝑥 = 0 ↔ 𝑥 = 0))
55	54	necon3bbid 2969	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (¬ 𝑥 = 0 ↔ 𝑥 ≠ 0))
56	55	pm5.74i 271	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 0) ↔ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0))
57	53, 56	sylib 218	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0))
58	57	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
59	16, 58	logcld 26535	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
60	13	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
61	6, 60, 58	redivcld 11969	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
62		dvrelog2b.4	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
63		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))
64		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))
65	21, 28, 18, 62, 63, 64	dvrelog3 42319	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
66		2cnd 12223	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
67	4	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
68	66, 67	logcld 26535	. . . 4 ⊢ (𝜑 → (log‘2) ∈ ℂ)
69		0red 11135	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
70		2rp 12910	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
71		loggt0b 26597	. . . . . . . . 9 ⊢ (2 ∈ ℝ⁺ → (0 < (log‘2) ↔ 1 < 2))
72	70, 71	ax-mp 5	. . . . . . . 8 ⊢ (0 < (log‘2) ↔ 1 < 2)
73	7, 72	mpbir 231	. . . . . . 7 ⊢ 0 < (log‘2)
74	73	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 < (log‘2))
75	69, 74	ltned 11269	. . . . 5 ⊢ (𝜑 → 0 ≠ (log‘2))
76	75	necomd 2987	. . . 4 ⊢ (𝜑 → (log‘2) ≠ 0)
77	51, 59, 61, 65, 68, 76	dvmptdivc 25925	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))))
78	3, 5	logcld 26535	. . . . . 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 11935	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((1 / 𝑥) / (log‘2)) = (1 / (𝑥 · (log‘2))))
81	80	mpteq2dva 5191	. . . 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 2742	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = 𝐺)
85	81, 84	eqtrd 2771	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))) = 𝐺)
86	77, 85	eqtrd 2771	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = 𝐺)
87	49, 86	eqtrd 2771	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 {csn 4580 {cpr 4582 class class class wbr 5098 ↦ cmpt 5179 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 · cmul 11031 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 / cdiv 11794 2c2 12200 ℝ⁺crp 12905 (,)cioo 13261 D cdv 25820 logclog 26519 log_b clogb 26730

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-log 26521 df-logb 26731

This theorem is referenced by: dvrelogpow2b 42322 aks4d1p1p6 42327

W3C validator