dvrelog3 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvrelog3		Structured version Visualization version GIF version

Theorem dvrelog3 41568

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 7442	. . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))))
4		reelprrecn 11238	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
6		rpcn 13024	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
7	6	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
8		rpne0 13030	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
9	8	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
10	7, 9	logcld 26524	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
11		1red 11253	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
12		rpre 13022	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
13	12	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
14	11, 13, 9	redivcld 12080	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
15		logf1o 26518	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
16		f1of 6844	. . . . . . . . . 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 13049	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ⁺
20		disjsn 4720	. . . . . . . . . . . 12 ⊢ ((ℝ⁺ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ⁺)
21	19, 20	mpbir 230	. . . . . . . . . . 11 ⊢ (ℝ⁺ ∩ {0}) = ∅
22		disjdif2 4483	. . . . . . . . . . 11 ⊢ ((ℝ⁺ ∩ {0}) = ∅ → (ℝ⁺ ∖ {0}) = ℝ⁺)
23	21, 22	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ⁺ ∖ {0}) = ℝ⁺
24		rpssre 13021	. . . . . . . . . . . 12 ⊢ ℝ⁺ ⊆ ℝ
25		ax-resscn 11203	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
26	24, 25	sstri 3991	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
27		ssdif 4140	. . . . . . . . . . 11 ⊢ (ℝ⁺ ⊆ ℂ → (ℝ⁺ ∖ {0}) ⊆ (ℂ ∖ {0}))
28	26, 27	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ⁺ ∖ {0}) ⊆ (ℂ ∖ {0})
29	23, 28	eqsstrri 4017	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ {0}))
31	18, 30	feqresmpt 6973	. . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
32	31	eqcomd 2734	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) = (log ↾ ℝ⁺))
33	32	oveq2d 7442	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ⁺)))
34		dvrelog 26591	. . . . . 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 2768	. . . 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 13405	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵)))
40	37, 38, 39	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵)))
41	40	biimpa 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵))
42	41	simp1d 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ)
43		0red 11255	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
44	43	rexrd 11302	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ^*)
45	37	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
46	42	rexrd 11302	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ^*)
47		dvrelog3.3	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐴)
48	47	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ 𝐴)
49	41	simp2d 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦)
50	44, 45, 46, 48, 49	xrlelttrd 13179	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 < 𝑦)
51	42, 50	jca 510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦))
52		elrp 13016	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦))
53	51, 52	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
54	53	ex 411	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ∈ ℝ⁺))
55	54	ssrdv 3988	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ⁺)
56		eqid 2728	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
57	56	tgioo2 24739	. . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
58		retop 24698	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
60		iooretop 24702	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
62		isopn3i 23006	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
63	59, 61, 62	syl2anc 582	. . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
64	5, 10, 14, 36, 55, 57, 56, 63	dvmptres2 25914	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
65	3, 64	eqtrd 2768	. 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 2734	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺)
69	65, 68	eqtrd 2768	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ∅c0 4326 {csn 4632 {cpr 4634 class class class wbr 5152 ↦ cmpt 5235 ran crn 5683 ↾ cres 5684 ⟶wf 6549 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 ℝ^*cxr 11285 < clt 11286 ≤ cle 11287 / cdiv 11909 ℝ⁺crp 13014 (,)cioo 13364 TopOpenctopn 17410 topGenctg 17426 ℂ_fldccnfld 21286 Topctop 22815 intcnt 22941 D cdv 25812 logclog 26508

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-cmp 23311 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510

This theorem is referenced by: dvrelog2b 41569

W3C validator