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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 12749 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ (ℂ ∖ {0})) | |
2 | 1 | adantl 482 | . . . . 5 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ (ℂ ∖ {0})) |
3 | rpcn 12740 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ) | |
4 | 3 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → 𝐵 ∈ ℂ) |
5 | rpne0 12746 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ≠ 0) | |
6 | 5 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → 𝐵 ≠ 0) |
7 | animorr 976 | . . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (𝐵 < 1 ∨ 1 < 𝐵)) | |
8 | rpre 12738 | . . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
9 | 1red 10976 | . . . . . . . . . . . 12 ⊢ (1 < 𝐵 → 1 ∈ ℝ) | |
10 | lttri2 11057 | . . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐵 ≠ 1 ↔ (𝐵 < 1 ∨ 1 < 𝐵))) | |
11 | 8, 9, 10 | syl2an 596 | . . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (𝐵 ≠ 1 ↔ (𝐵 < 1 ∨ 1 < 𝐵))) |
12 | 7, 11 | mpbird 256 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → 𝐵 ≠ 1) |
13 | 4, 6, 12 | 3jca 1127 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1)) |
14 | logbmpt 25938 | . . . . . . . . 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 5814 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → dom (curry logb ‘𝐵) = dom (𝑥 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑥) / (log‘𝐵)))) |
17 | ovexd 7310 | . . . . . . . . 9 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((log‘𝑥) / (log‘𝐵)) ∈ V) | |
18 | 17 | ralrimiva 3103 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → ∀𝑥 ∈ (ℂ ∖ {0})((log‘𝑥) / (log‘𝐵)) ∈ V) |
19 | dmmptg 6145 | . . . . . . . 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 2778 | . . . . . 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 2842 | . . . 4 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ dom (curry logb ‘𝐵)) |
24 | logbfval 25940 | . . . . . 6 ⊢ (((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ∧ 𝑥 ∈ (ℂ ∖ {0})) → ((curry logb ‘𝐵)‘𝑥) = (𝐵 logb 𝑥)) | |
25 | 13, 1, 24 | syl2an 596 | . . . . 5 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → ((curry logb ‘𝐵)‘𝑥) = (𝐵 logb 𝑥)) |
26 | simpll 764 | . . . . . . 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 1127 | . . . . . 6 ⊢ (((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) ∧ 𝑥 ∈ ℝ+) → (𝐵 ∈ ℝ+ ∧ 𝑥 ∈ ℝ+ ∧ 𝐵 ≠ 1)) |
30 | relogbcl 25923 | . . . . . 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 2839 | . . . 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 3103 | . 2 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → ∀𝑥 ∈ ℝ+ (𝑥 ∈ dom (curry logb ‘𝐵) ∧ ((curry logb ‘𝐵)‘𝑥) ∈ ℝ)) |
35 | logbf 25939 | . . . 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 6603 | . . 3 ⊢ ((curry logb ‘𝐵):(ℂ ∖ {0})⟶ℂ → Fun (curry logb ‘𝐵)) | |
38 | ffvresb 6998 | . . 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 256 | 1 ⊢ ((𝐵 ∈ ℝ+ ∧ 1 < 𝐵) → ((curry logb ‘𝐵) ↾ ℝ+):ℝ+⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3432 ∖ cdif 3884 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ↾ cres 5591 Fun wfun 6427 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 curry ccur 8081 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 < clt 11009 / cdiv 11632 ℝ+crp 12730 logclog 25710 logb clogb 25914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-cur 8083 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-logb 25915 |
This theorem is referenced by: (None) |
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