dvrelog3 - Mathbox for metakunt

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Theorem dvrelog3 42813

Description: The derivative of the logarithm on an open interval. (Contributed by metakunt, 11-Aug-2024.)

**Hypotheses**
Ref	Expression
dvrelog3.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
dvrelog3.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
dvrelog3.3	⊢ (𝜑 → 0 ≤ 𝐴)
dvrelog3.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)
dvrelog3.5	⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))
dvrelog3.6	⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))

**Assertion**
Ref	Expression
dvrelog3	⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

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

Proof of Theorem dvrelog3 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvrelog3.5	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)))
3	2	oveq2d 7428	. . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))))
4		reelprrecn 11193	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
6		rpcn 13028	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
7	6	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
8		rpne0 13034	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
9	8	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
10	7, 9	logcld 26716	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
11		1red 11210	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
12		rpre 13026	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
13	12	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
14	11, 13, 9	redivcld 12044	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
15		logf1o 26710	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
16		f1of 6822	. . . . . . . . . 10 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
17	15, 16	ax-mp 5	. . . . . . . . 9 ⊢ log:(ℂ ∖ {0})⟶ran log
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → log:(ℂ ∖ {0})⟶ran log)
19		0nrp 13054	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ⁺
20		disjsn 4678	. . . . . . . . . . . 12 ⊢ ((ℝ⁺ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ⁺)
21	19, 20	mpbir 234	. . . . . . . . . . 11 ⊢ (ℝ⁺ ∩ {0}) = ∅
22		disjdif2 4442	. . . . . . . . . . 11 ⊢ ((ℝ⁺ ∩ {0}) = ∅ → (ℝ⁺ ∖ {0}) = ℝ⁺)
23	21, 22	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ⁺ ∖ {0}) = ℝ⁺
24		rpssre 13025	. . . . . . . . . . . 12 ⊢ ℝ⁺ ⊆ ℝ
25		ax-resscn 11158	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
26	24, 25	sstri 3947	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
27		ssdif 4099	. . . . . . . . . . 11 ⊢ (ℝ⁺ ⊆ ℂ → (ℝ⁺ ∖ {0}) ⊆ (ℂ ∖ {0}))
28	26, 27	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ⁺ ∖ {0}) ⊆ (ℂ ∖ {0})
29	23, 28	eqsstrri 3985	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ {0}))
31	18, 30	feqresmpt 6952	. . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
32	31	eqcomd 2769	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) = (log ↾ ℝ⁺))
33	32	oveq2d 7428	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ⁺)))
34		dvrelog 26783	. . . . . 6 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
35	34	a1i 11	. . . . 5 ⊢ (𝜑 → (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
36	33, 35	eqtrd 2798	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
37		dvrelog3.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
38		dvrelog3.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
39		elioo2 13414	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵)))
40	37, 38, 39	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵)))
41	40	biimpa 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵))
42	41	simp1d 1160	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ)
43		0red 11212	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
44	43	rexrd 11260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ^*)
45	37	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
46	42	rexrd 11260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ^*)
47		dvrelog3.3	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐴)
48	47	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ 𝐴)
49	41	simp2d 1161	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦)
50	44, 45, 46, 48, 49	xrlelttrd 13186	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 < 𝑦)
51	42, 50	jca 520	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦))
52		elrp 13019	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦))
53	51, 52	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
54	53	ex 417	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ∈ ℝ⁺))
55	54	ssrdv 3944	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ⁺)
56		tgioo4 24943	. . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
57		eqid 2763	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
58		retop 24899	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
60		iooretop 24903	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
62		isopn3i 23220	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
63	59, 61, 62	syl2anc 595	. . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
64	5, 10, 14, 36, 55, 56, 57, 63	dvmptres2 26102	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
65	3, 64	eqtrd 2798	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
66		dvrelog3.6	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))
67	66	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
68	67	eqcomd 2769	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺)
69	65, 68	eqtrd 2798	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3903 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 {csn 4590 {cpr 4592 class class class wbr 5110 ↦ cmpt 5193 ran crn 5664 ↾ cres 5665 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 / cdiv 11872 ℝ⁺crp 13017 (,)cioo 13373 TopOpenctopn 17475 topGenctg 17491 ℂ_fldccnfld 21503 Topctop 23031 intcnt 23155 D cdv 26003 logclog 26700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 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 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-ef 16122 df-sin 16124 df-cos 16125 df-pi 16127 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-mulg 19135 df-cntz 19388 df-cmn 19853 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-fg 21501 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-lp 23274 df-perf 23275 df-cn 23365 df-cnp 23366 df-haus 23453 df-cmp 23525 df-tx 23700 df-hmeo 23893 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-xms 24458 df-ms 24459 df-tms 24460 df-cncf 25018 df-limc 26006 df-dv 26007 df-log 26702

This theorem is referenced by: dvrelog2b 42814

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