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Theorem dvrelog2 42097

Description: The derivative of the logarithm, ftc2 25973 version. (Contributed by metakunt, 11-Aug-2024.)

**Hypotheses**
Ref	Expression
dvrelog2.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
dvrelog2.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
dvrelog2.3	⊢ (𝜑 → 0 < 𝐴)
dvrelog2.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)
dvrelog2.5	⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))
dvrelog2.6	⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))

**Assertion**
Ref	Expression
dvrelog2	⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

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

Proof of Theorem dvrelog2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvrelog2.5	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥)))
3	2	oveq2d 7357	. . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))))
4		reelprrecn 11093	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
6		rpssre 12893	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
7		ax-resscn 11058	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
8	6, 7	sstri 3939	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℂ
9	8	sseli 3925	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
10	9	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
11		rpne0 12902	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
12	11	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
13	10, 12	logcld 26501	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
14		1red 11108	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 1 ∈ ℝ)
15	6	sseli 3925	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
16	14, 15, 11	redivcld 11944	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
17	16	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
18		logf1o 26495	. . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
19		f1of 6758	. . . . . . . . . 10 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log)
20	18, 19	ax-mp 5	. . . . . . . . 9 ⊢ log:(ℂ ∖ {0})⟶ran log
21	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → log:(ℂ ∖ {0})⟶ran log)
22		0nrp 12922	. . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ⁺
23		disjsn 4659	. . . . . . . . . . . 12 ⊢ ((ℝ⁺ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ⁺)
24	22, 23	mpbir 231	. . . . . . . . . . 11 ⊢ (ℝ⁺ ∩ {0}) = ∅
25		disjdif2 4425	. . . . . . . . . . 11 ⊢ ((ℝ⁺ ∩ {0}) = ∅ → (ℝ⁺ ∖ {0}) = ℝ⁺)
26	24, 25	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ⁺ ∖ {0}) = ℝ⁺
27		ssdif 4089	. . . . . . . . . . 11 ⊢ (ℝ⁺ ⊆ ℂ → (ℝ⁺ ∖ {0}) ⊆ (ℂ ∖ {0}))
28	8, 27	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ⁺ ∖ {0}) ⊆ (ℂ ∖ {0})
29	26, 28	eqsstrri 3977	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ (ℂ ∖ {0})
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ (ℂ ∖ {0}))
31	21, 30	feqresmpt 6886	. . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
32	31	eqcomd 2737	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)) = (log ↾ ℝ⁺))
33	32	oveq2d 7357	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ⁺)))
34		dvrelog 26568	. . . . . 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 2766	. . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
37		dvrelog2.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
38		dvrelog2.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
39		elicc2 13306	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
40	37, 38, 39	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)))
41	40	biimpa 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))
42	41	simp1d 1142	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ)
43		0red 11110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 0 ∈ ℝ)
44	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ)
45		dvrelog2.3	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴)
46	45	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 0 < 𝐴)
47	41	simp2d 1143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑦)
48	43, 44, 42, 46, 47	ltletrd 11268	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 0 < 𝑦)
49	42, 48	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦))
50		elrp 12887	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦))
51	49, 50	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ⁺)
52	51	ex 412	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ⁺))
53	52	ssrdv 3935	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ⁺)
54		tgioo4 24715	. . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
55		eqid 2731	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
56		iccntr 24732	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
57	37, 38, 56	syl2anc 584	. . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
58	5, 13, 17, 36, 53, 54, 55, 57	dvmptres2 25888	. . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
59	3, 58	eqtrd 2766	. 2 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
60		dvrelog2.6	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))
61	60	a1i 11	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)))
62	61	eqcomd 2737	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺)
63	59, 62	eqtrd 2766	1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3894 ∩ cin 3896 ⊆ wss 3897 ∅c0 4278 {csn 4571 {cpr 4573 class class class wbr 5086 ↦ cmpt 5167 ran crn 5612 ↾ cres 5613 ⟶wf 6472 –1-1-onto→wf1o 6475 ‘cfv 6476 (class class class)co 7341 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 < clt 11141 ≤ cle 11142 / cdiv 11769 ℝ⁺crp 12885 (,)cioo 13240 [,]cicc 13243 TopOpenctopn 17320 topGenctg 17336 ℂ_fldccnfld 21286 intcnt 22927 D cdv 25786 logclog 26485

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-mod 13769 df-seq 13904 df-exp 13964 df-fac 14176 df-bc 14205 df-hash 14233 df-shft 14969 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-limsup 15373 df-clim 15390 df-rlim 15391 df-sum 15589 df-ef 15969 df-sin 15971 df-cos 15972 df-pi 15974 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 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 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-perf 23047 df-cn 23137 df-cnp 23138 df-haus 23225 df-cmp 23297 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-limc 25789 df-dv 25790 df-log 26487

This theorem is referenced by: (None)

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