dvrelog2b - Mathbox for metakunt

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Theorem dvrelog2b 43116

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 12421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ)
4		2ne0 12449	. . . . . . . . 9 ⊢ 2 ≠ 0
5	4	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0)
6		1red 11309	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ)
7		1lt2 12515	. . . . . . . . . . 11 ⊢ 1 < 2
8	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2)
9	6, 8	ltned 11446	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2)
10	9	necomd 3011	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1)
11	5, 10	nelprd 4618	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0, 1})
12	3, 11	eldifd 3910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0, 1}))
13		elioore 13506	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
14		recn 11290	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
15	13, 14	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℂ)
16	15	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
17		elsni 4601	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {0} → 𝑥 = 0)
18		dvrelog2b.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 ≤ 𝐴)
19		0xr 11356	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ ℝ^*
20	19	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℝ^*)
21		dvrelog2b.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		xrlenlt 11374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
23	20, 21, 22	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 ≤ 𝐴 ↔ ¬ 𝐴 < 0))
24	18, 23	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝐴 < 0)
25	24	orcd 887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (¬ 𝐴 < 0 ∨ ¬ 0 < 𝐵))
26		ianor 997	. . . . . . . . . . . . . . . . . 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 13534	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 0 ∈ ℝ^*) → (0 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 0 ∧ 0 < 𝐵)))
30	21, 28, 20, 29	syl3anc 1398	. . . . . . . . . . . . . . . . . 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 26	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 ∈ (𝐴(,)𝐵) → ¬ 0 ∈ (𝐴(,)𝐵)))
34	33	imp 412	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ (𝐴(,)𝐵))
35	34	pm2.01da 811	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))
36	35	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 = 0) → ¬ 0 ∈ (𝐴(,)𝐵))
37		eleq1 2849	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 ∈ (𝐴(,)𝐵) ↔ 0 ∈ (𝐴(,)𝐵)))
38	37	adantl 487	. . . . . . . . . . . 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 605	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {0}) → ¬ 𝑥 ∈ (𝐴(,)𝐵))
41	40	ex 418	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ {0} → ¬ 𝑥 ∈ (𝐴(,)𝐵)))
42	41	con2d 135	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 ∈ {0}))
43	42	imp 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0})
44	16, 43	eldifd 3910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖ {0}))
45		logbval 27094	. . . . . 6 ⊢ ((2 ∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
46	12, 44, 45	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 log_b 𝑥) = ((log‘𝑥) / (log‘2)))
47	46	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 log_b 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
48	2, 47	eqtrd 2796	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2))))
49	48	oveq2d 7436	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))))
50		reelprrecn 11292	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
51	50	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
52	39	ex 418	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 0 → ¬ 𝑥 ∈ (𝐴(,)𝐵)))
53	52	con2d 135	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) → ¬ 𝑥 = 0))
54		biidd 265	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → (𝑥 = 0 ↔ 𝑥 = 0))
55	54	necon3bbid 2993	. . . . . . . 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 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)
59	16, 58	logcld 26898	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ)
60	13	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ)
61	6, 60, 58	redivcld 12145	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / 𝑥) ∈ ℝ)
62		dvrelog2b.4	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
63		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))
64		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))
65	21, 28, 18, 62, 63, 64	dvrelog3 43115	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
66		2cnd 12421	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
67	4	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ≠ 0)
68	66, 67	logcld 26898	. . . 4 ⊢ (𝜑 → (log‘2) ∈ ℂ)
69		0red 11311	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ)
70		2rp 13125	. . . . . . . . 9 ⊢ 2 ∈ ℝ⁺
71		loggt0b 26960	. . . . . . . . 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 11446	. . . . 5 ⊢ (𝜑 → 0 ≠ (log‘2))
76	75	necomd 3011	. . . 4 ⊢ (𝜑 → (log‘2) ≠ 0)
77	51, 59, 61, 65, 68, 76	dvmptdivc 26285	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))))
78	3, 5	logcld 26898	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈ ℂ)
79	76	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠ 0)
80	16, 78, 58, 79	recdiv2d 12111	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((1 / 𝑥) / (log‘2)) = (1 / (𝑥 · (log‘2))))
81	80	mpteq2dva 5198	. . . 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 2767	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = 𝐺)
85	81, 84	eqtrd 2796	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((1 / 𝑥) / (log‘2))) = 𝐺)
86	77, 85	eqtrd 2796	. 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((log‘𝑥) / (log‘2)))) = 𝐺)
87	49, 86	eqtrd 2796	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∖ cdif 3896 {csn 4584 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 · cmul 11205 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 / cdiv 11973 2c2 12397 ℝ⁺crp 13120 (,)cioo 13476 D cdv 26183 logclog 26882 log_b clogb 27092

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-log 26884 df-logb 27093

This theorem is used by: dvrelogpow2b 43118 aks4d1p1p6 43123

W3C validator