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Mirrors > Home > MPE Home > Th. List > relogbf | Structured version Visualization version GIF version |
Description: The general logarithm to a real base greater than 1 regarded as function restricted to the positive integers. Property in [Cohen4] p. 349. (Contributed by AV, 12-Jun-2020.) |
Ref | Expression |
---|---|
relogbf | ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → ((curry logb ‘𝐵) ↾ ℝ+):ℝ+⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpcndif0 12396 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ (ℂ ∖ {0})) | |
2 | 1 | adantl 482 | . . . . 5 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ (ℂ ∖ {0})) |
3 | rpcn 12387 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ) | |
4 | 3 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → 𝐵 ∈ ℂ) |
5 | rpne0 12393 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ≠ 0) | |
6 | 5 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → 𝐵 ≠ 0) |
7 | animorr 972 | . . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (𝐵 < 1 ∨ 1 < 𝐵)) | |
8 | rpre 12385 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
9 | 1red 10630 | . . . . . . . . . . . 12 ⊢ (1 < 𝐵 → 1 ∈ ℝ) | |
10 | lttri2 10711 | . . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐵 ≠ 1 ↔ (𝐵 < 1 ∨ 1 < 𝐵))) | |
11 | 8, 9, 10 | syl2an 595 | . . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (𝐵 ≠ 1 ↔ (𝐵 < 1 ∨ 1 < 𝐵))) |
12 | 7, 11 | mpbird 258 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → 𝐵 ≠ 1) |
13 | 4, 6, 12 | 3jca 1120 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) |
14 | logbmpt 25293 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵) = (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝐵)))) | |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (curry logb ‘𝐵) = (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝐵)))) |
16 | 15 | dmeqd 5767 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → dom (curry logb ‘𝐵) = dom (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝐵)))) |
17 | ovexd 7180 | . . . . . . . . 9 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((log‘𝑥) / (log‘𝐵)) ∈ V) | |
18 | 17 | ralrimiva 3179 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → ∀𝑥 ∈ (ℂ ∖ {0})((log‘𝑥) / (log‘𝐵)) ∈ V) |
19 | dmmptg 6089 | . . . . . . . 8 ⊢ (∀𝑥 ∈ (ℂ ∖ {0})((log‘𝑥) / (log‘𝐵)) ∈ V → dom (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝐵))) = (ℂ ∖ {0})) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → dom (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝐵))) = (ℂ ∖ {0})) |
21 | 16, 20 | eqtrd 2853 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → dom (curry logb ‘𝐵) = (ℂ ∖ {0})) |
22 | 21 | adantr 481 | . . . . 5 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → dom (curry logb ‘𝐵) = (ℂ ∖ {0})) |
23 | 2, 22 | eleqtrrd 2913 | . . . 4 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ dom (curry logb ‘𝐵)) |
24 | logbfval 25295 | . . . . . 6 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((curry logb ‘𝐵)‘𝑥) = (𝐵 logb 𝑥)) | |
25 | 13, 1, 24 | syl2an 595 | . . . . 5 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → ((curry logb ‘𝐵)‘𝑥) = (𝐵 logb 𝑥)) |
26 | simpll 763 | . . . . . . 7 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝐵 ∈ ℝ+) | |
27 | simpr 485 | . . . . . . 7 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+) | |
28 | 12 | adantr 481 | . . . . . . 7 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝐵 ≠ 1) |
29 | 26, 27, 28 | 3jca 1120 | . . . . . 6 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → (𝐵 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+ ∧ 𝐵 ≠ 1)) |
30 | relogbcl 25278 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+ ∧ 𝐵 ≠ 1) → (𝐵 logb 𝑥) ∈ ℝ) | |
31 | 29, 30 | syl 17 | . . . . 5 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → (𝐵 logb 𝑥) ∈ ℝ) |
32 | 25, 31 | eqeltrd 2910 | . . . 4 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → ((curry logb ‘𝐵)‘𝑥) ∈ ℝ) |
33 | 23, 32 | jca 512 | . . 3 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ dom (curry logb ‘𝐵) ∧ ((curry logb ‘𝐵)‘𝑥) ∈ ℝ)) |
34 | 33 | ralrimiva 3179 | . 2 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → ∀𝑥 ∈ ℝ+ (𝑥 ∈ dom (curry logb ‘𝐵) ∧ ((curry logb ‘𝐵)‘𝑥) ∈ ℝ)) |
35 | logbf 25294 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (curry logb ‘𝐵):(ℂ ∖ {0})⟶ℂ) | |
36 | 13, 35 | syl 17 | . . 3 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (curry logb ‘𝐵):(ℂ ∖ {0})⟶ℂ) |
37 | ffun 6510 | . . 3 ⊢ ((curry logb ‘𝐵):(ℂ ∖ {0})⟶ℂ → Fun (curry logb ‘𝐵)) | |
38 | ffvresb 6880 | . . 3 ⊢ (Fun (curry logb ‘𝐵) → (((curry logb ‘𝐵) ↾ ℝ+):ℝ+⟶ℝ ↔ ∀𝑥 ∈ ℝ+ (𝑥 ∈ dom (curry logb ‘𝐵) ∧ ((curry logb ‘𝐵)‘𝑥) ∈ ℝ))) | |
39 | 36, 37, 38 | 3syl 18 | . 2 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (((curry logb ‘𝐵) ↾ ℝ+):ℝ+⟶ℝ ↔ ∀𝑥 ∈ ℝ+ (𝑥 ∈ dom (curry logb ‘𝐵) ∧ ((curry logb ‘𝐵)‘𝑥) ∈ ℝ))) |
40 | 34, 39 | mpbird 258 | 1 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → ((curry logb ‘𝐵) ↾ ℝ+):ℝ+⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 Vcvv 3492 ∖ cdif 3930 {csn 4557 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 curry ccur 7920 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 < clt 10663 / cdiv 11285 ℝ+crp 12377 logclog 25065 logb clogb 25269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-cur 7922 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-logb 25270 |
This theorem is referenced by: (None) |
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