dvrelog3 - Mathbox for metakunt

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Theorem dvrelog3 43095

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 7434	. . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))))
4		reelprrecn 11285	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
6		rpcn 13124	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
7	6	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
8		rpne0 13130	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
9	8	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
10	7, 9	logcld 26891	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
11		1red 11302	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
12		rpre 13122	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
13	12	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
14	11, 13, 9	redivcld 12138	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
15		logf1o 26885	. . . . . . . . . 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 13150	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ⁺
20		disjsn 4672	. . . . . . . . . . . 12 ⊢ ((ℝ⁺ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ⁺)
21	19, 20	mpbir 234	. . . . . . . . . . 11 ⊢ (ℝ⁺ ∩ {0}) = ∅
22		disjdif2 4436	. . . . . . . . . . 11 ⊢ ((ℝ⁺ ∩ {0}) = ∅ → (ℝ⁺ ∖ {0}) = ℝ⁺)
23	21, 22	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ⁺ ∖ {0}) = ℝ⁺
24		rpssre 13121	. . . . . . . . . . . 12 ⊢ ℝ⁺ ⊆ ℝ
25		ax-resscn 11250	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
26	24, 25	sstri 3940	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ ℂ
27		ssdif 4091	. . . . . . . . . . 11 ⊢ (ℝ⁺ ⊆ ℂ → (ℝ⁺ ∖ {0}) ⊆ (ℂ ∖ {0}))
28	26, 27	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ⁺ ∖ {0}) ⊆ (ℂ ∖ {0})
29	23, 28	eqsstrri 3978	. . . . . . . . 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 2767	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) = (log ↾ ℝ⁺))
33	32	oveq2d 7434	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ⁺)))
34		dvrelog 26958	. . . . . 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 2796	. . . 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 13510	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵)))
40	37, 38, 39	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵)))
41	40	biimpa 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵))
42	41	simp1d 1160	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ)
43		0red 11304	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ)
44	43	rexrd 11352	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ^*)
45	37	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
46	42	rexrd 11352	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ^*)
47		dvrelog3.3	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐴)
48	47	adantr 486	. . . . . . . . 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 13282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 < 𝑦)
51	42, 50	jca 521	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦))
52		elrp 13115	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦))
53	51, 52	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ⁺)
54	53	ex 418	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ∈ ℝ⁺))
55	54	ssrdv 3937	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ⁺)
56		tgioo4 25117	. . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
57		eqid 2761	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
58		retop 25073	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top)
60		iooretop 25077	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
62		isopn3i 23393	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
63	59, 61, 62	syl2anc 596	. . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵))
64	5, 10, 14, 36, 55, 56, 57, 63	dvmptres2 26275	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
65	3, 64	eqtrd 2796	. 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 2767	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺)
69	65, 68	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 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 ran crn 5652 ↾ cres 5653 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ℝcr 11192 0cc0 11193 1c1 11194 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 / cdiv 11966 ℝ⁺crp 13113 (,)cioo 13469 TopOpenctopn 17585 topGenctg 17601 ℂ_fldccnfld 21671 Topctop 23204 intcnt 23328 D cdv 26176 logclog 26875

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 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

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 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ioc 13474 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-mod 14003 df-seq 14138 df-exp 14198 df-fac 14411 df-bc 14440 df-hash 14468 df-shft 15213 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-limsup 15631 df-clim 15648 df-rlim 15649 df-sum 15847 df-ef 16226 df-sin 16228 df-cos 16229 df-pi 16231 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cn 23538 df-cnp 23539 df-haus 23626 df-cmp 23698 df-tx 23874 df-hmeo 24067 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-limc 26179 df-dv 26180 df-log 26877

This theorem is used by: dvrelog2b 43096

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