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Mirrors > Home > MPE Home > Th. List > relogcl | Structured version Visualization version GIF version |
Description: Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
Ref | Expression |
---|---|
relogcl | ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6736 | . 2 ⊢ (𝐴 ∈ ℝ+ → ((log ↾ ℝ+)‘𝐴) = (log‘𝐴)) | |
2 | relogf1o 25455 | . . . 4 ⊢ (log ↾ ℝ+):ℝ+–1-1-onto→ℝ | |
3 | f1of 6661 | . . . 4 ⊢ ((log ↾ ℝ+):ℝ+–1-1-onto→ℝ → (log ↾ ℝ+):ℝ+⟶ℝ) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (log ↾ ℝ+):ℝ+⟶ℝ |
5 | 4 | ffvelrni 6903 | . 2 ⊢ (𝐴 ∈ ℝ+ → ((log ↾ ℝ+)‘𝐴) ∈ ℝ) |
6 | 1, 5 | eqeltrrd 2839 | 1 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ↾ cres 5553 ⟶wf 6376 –1-1-onto→wf1o 6379 ‘cfv 6380 ℝcr 10728 ℝ+crp 12586 logclog 25443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629 df-sin 15631 df-cos 15632 df-pi 15634 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764 df-log 25445 |
This theorem is referenced by: logneg 25476 lognegb 25478 relogoprlem 25479 reexplog 25483 relogexp 25484 logfac 25489 logleb 25491 rplogcl 25492 logmul2 25504 logdiv2 25505 abslogle 25506 logdivlti 25508 logdivlt 25509 logdivle 25510 relogcld 25511 advlog 25542 advlogexp 25543 logccv 25551 logcxp 25557 rpcxpcl 25564 cxpmul 25576 abscxp 25580 cxple2 25585 logsqrt 25592 dvcxp1 25626 dvcxp2 25627 loglesqrt 25644 relogbcl 25656 relogbmul 25660 logbgt0b 25676 log2ub 25832 log2le1 25833 birthday 25837 cxploglim 25860 cxploglim2 25861 amgmlem 25872 logdifbnd 25876 emcllem7 25884 emre 25888 emgt0 25889 harmonicbnd3 25890 harmoniclbnd 25891 harmonicbnd4 25893 relgamcl 25944 cht2 26054 chtleppi 26091 chtublem 26092 chtub 26093 logfacubnd 26102 logfaclbnd 26103 logfacbnd3 26104 logfacrlim 26105 logexprlim 26106 efexple 26162 bposlem6 26170 bposlem7 26171 bposlem8 26172 bposlem9 26173 chebbnd1lem3 26352 chebbnd1 26353 chto1ub 26357 vmadivsum 26363 rpvmasumlem 26368 dchrvmasumlem2 26379 dchrvmasumlema 26381 dchrvmasumiflem1 26382 dchrvmasumiflem2 26383 dchrisum0fno1 26392 rpvmasum2 26393 dchrisum0re 26394 rpvmasum 26407 rplogsum 26408 dirith2 26409 logdivsum 26414 mulog2sumlem2 26416 mulog2sumlem3 26417 logsqvma 26423 log2sumbnd 26425 selberglem1 26426 selberglem2 26427 selberglem3 26428 selberg 26429 selberg2lem 26431 selberg2 26432 pntrsumo1 26446 selbergr 26449 pntrlog2bndlem4 26461 pntibndlem3 26473 xrge0iifiso 31599 logdivsqrle 32342 hgt750lem 32343 hgt750lemb 32348 reglogcl 40415 reglogltb 40416 reglogleb 40417 reglogmul 40418 reglogexp 40419 reglogbas 40420 reglog1 40421 stirlinglem12 43301 stirlinglem13 43302 stirlinglem14 43303 lighneallem2 44731 logbge0b 45582 logblt1b 45583 |
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