dvrelog2b - Mathbox for metakunt

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Theorem dvrelog2b 40920

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 12287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2ne0 12313	. . . . . . . . 9 ⊢ 2 ≠ 0
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
6		1red 11212	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
7		1lt2 12380	. . . . . . . . . . 11 ⊢ 1 < 2
8	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
9	6, 8	ltned 11347	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
10	9	necomd 2997	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
11	5, 10	nelprd 4659	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
12	3, 11	eldifd 3959	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
13		elioore 13351	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
14		recn 11197	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℂ)
16	15	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
17		elsni 4645	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
18		dvrelog2b.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 ≤ 𝐴)
19		0xr 11258	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℝ^*
20	19	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℝ^*)
21		dvrelog2b.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		xrlenlt 11276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
23	20, 21, 22	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
24	18, 23	mpbid 231	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝐴 < 0)
25	24	orcd 872	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵))
26		ianor 981	. . . . . . . . . . . . . . . . . 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 13378	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵)))
30	21, 28, 20, 29	syl3anc 1372	. . . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ (𝐴(,)𝐵))
35	34	pm2.01da 798	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
36	35	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
37		eleq1 2822	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵)))
38	37	adantl 483	. . . . . . . . . . . 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 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {0}) → ¬ 𝑥 ∈ (𝐴(,)𝐵))
41	40	ex 414	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ {0} → ¬ 𝑥 ∈ (𝐴(,)𝐵)))
42	41	con2d 134	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 ∈ {0}))
43	42	imp 408	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0})
44	16, 43	eldifd 3959	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
45		logbval 26261	. . . . . 6 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
46	12, 44, 45	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
47	46	mpteq2dva 5248	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
48	2, 47	eqtrd 2773	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
49	48	oveq2d 7422	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))))
50		reelprrecn 11199	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
51	50	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
52	39	ex 414	. . . . . . . 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 2979	. . . . . . . 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 217	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ≠ 0))
58	57	imp 408	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
59	16, 58	logcld 26071	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
60	13	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
61	6, 60, 58	redivcld 12039	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
62		dvrelog2b.4	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
63		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))
64		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))
65	21, 28, 18, 62, 63, 64	dvrelog3 40919	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
66		2cnd 12287	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
67	4	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
68	66, 67	logcld 26071	. . . 4 ⊢ (𝜑 → (log‘2) ∈ ℂ)
69		0red 11214	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
70		2rp 12976	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
71		loggt0b 26132	. . . . . . . . 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 11347	. . . . 5 ⊢ (𝜑 → 0 ≠ (log‘2))
76	75	necomd 2997	. . . 4 ⊢ (𝜑 → (log‘2) ≠ 0)
77	51, 59, 61, 65, 68, 76	dvmptdivc 25474	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))))
78	3, 5	logcld 26071	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
79	76	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
80	16, 78, 58, 79	recdiv2d 12005	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((1 / 𝑥) / (log‘2)) = (1 / (𝑥 · (log‘2))))
81	80	mpteq2dva 5248	. . . 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 2739	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = 𝐺)
85	81, 84	eqtrd 2773	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))) = 𝐺)
86	77, 85	eqtrd 2773	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = 𝐺)
87	49, 86	eqtrd 2773	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3945 {csn 4628 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 · cmul 11112 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 / cdiv 11868 2c2 12264 ℝ⁺crp 12971 (,)cioo 13321 D cdv 25372 logclog 26055 log_b clogb 26259

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-haus 22811 df-cmp 22883 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-xms 23818 df-ms 23819 df-tms 23820 df-cncf 24386 df-limc 25375 df-dv 25376 df-log 26057 df-logb 26260

This theorem is referenced by: dvrelogpow2b 40922 aks4d1p1p6 40927

W3C validator