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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dvrelog3 | Structured version Visualization version GIF version | ||
| Description: The derivative of the logarithm on an open interval. (Contributed by metakunt, 11-Aug-2024.) |
| 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)) |
| Ref | Expression |
|---|---|
| dvrelog3 | ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |
| 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 7447 | . . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)))) |
| 4 | reelprrecn 11247 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
| 5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
| 6 | rpcn 13045 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) | |
| 7 | 6 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ) |
| 8 | rpne0 13051 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
| 9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
| 10 | 7, 9 | logcld 26612 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
| 11 | 1red 11262 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ) | |
| 12 | rpre 13043 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
| 13 | 12 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ) |
| 14 | 11, 13, 9 | redivcld 12095 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ) |
| 15 | logf1o 26606 | . . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
| 16 | f1of 6848 | . . . . . . . . . 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 13070 | . . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ+ | |
| 20 | disjsn 4711 | . . . . . . . . . . . 12 ⊢ ((ℝ+ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ+) | |
| 21 | 19, 20 | mpbir 231 | . . . . . . . . . . 11 ⊢ (ℝ+ ∩ {0}) = ∅ |
| 22 | disjdif2 4480 | . . . . . . . . . . 11 ⊢ ((ℝ+ ∩ {0}) = ∅ → (ℝ+ ∖ {0}) = ℝ+) | |
| 23 | 21, 22 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) = ℝ+ |
| 24 | rpssre 13042 | . . . . . . . . . . . 12 ⊢ ℝ+ ⊆ ℝ | |
| 25 | ax-resscn 11212 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
| 26 | 24, 25 | sstri 3993 | . . . . . . . . . . 11 ⊢ ℝ+ ⊆ ℂ |
| 27 | ssdif 4144 | . . . . . . . . . . 11 ⊢ (ℝ+ ⊆ ℂ → (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0})) | |
| 28 | 26, 27 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0}) |
| 29 | 23, 28 | eqsstrri 4031 | . . . . . . . . 9 ⊢ ℝ+ ⊆ (ℂ ∖ {0}) |
| 30 | 29 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ+ ⊆ (ℂ ∖ {0})) |
| 31 | 18, 30 | feqresmpt 6978 | . . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
| 32 | 31 | eqcomd 2743 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) = (log ↾ ℝ+)) |
| 33 | 32 | oveq2d 7447 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ+))) |
| 34 | dvrelog 26679 | . . . . . 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 2777 | . . . 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 13428 | . . . . . . . . . . 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 1143 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ) |
| 43 | 0red 11264 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ) | |
| 44 | 43 | rexrd 11311 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ*) |
| 45 | 37 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ*) |
| 46 | 42 | rexrd 11311 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ*) |
| 47 | dvrelog3.3 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐴) | |
| 48 | 47 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ 𝐴) |
| 49 | 41 | simp2d 1144 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦) |
| 50 | 44, 45, 46, 48, 49 | xrlelttrd 13202 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 < 𝑦) |
| 51 | 42, 50 | jca 511 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦)) |
| 52 | elrp 13036 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ+ ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) | |
| 53 | 51, 52 | sylibr 234 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ+) |
| 54 | 53 | ex 412 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ∈ ℝ+)) |
| 55 | 54 | ssrdv 3989 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ+) |
| 56 | tgioo4 24826 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) | |
| 57 | eqid 2737 | . . . 4 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
| 58 | retop 24782 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
| 59 | 58 | a1i 11 | . . . . 5 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
| 60 | iooretop 24786 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
| 61 | 60 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,))) |
| 62 | isopn3i 23090 | . . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)) | |
| 63 | 59, 61, 62 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)) |
| 64 | 5, 10, 14, 36, 55, 56, 57, 63 | dvmptres2 26000 | . . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
| 65 | 3, 64 | eqtrd 2777 | . 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 2743 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺) |
| 69 | 65, 68 | eqtrd 2777 | 1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 / cdiv 11920 ℝ+crp 13034 (,)cioo 13387 TopOpenctopn 17466 topGenctg 17482 ℂfldccnfld 21364 Topctop 22899 intcnt 23025 D cdv 25898 logclog 26596 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 |
| This theorem is referenced by: dvrelog2b 42067 |
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