Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rplogcld | Structured version Visualization version GIF version |
Description: Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
relogefd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rplogcld.2 | ⊢ (𝜑 → 1 < 𝐴) |
Ref | Expression |
---|---|
rplogcld | ⊢ (𝜑 → (log‘𝐴) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relogefd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | rplogcld.2 | . 2 ⊢ (𝜑 → 1 < 𝐴) | |
3 | rplogcl 25757 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (log‘𝐴) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 ℝcr 10871 1c1 10873 < clt 11010 ℝ+crp 12729 logclog 25708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 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 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ioc 13083 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-fac 13986 df-bc 14015 df-hash 14043 df-shft 14776 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-limsup 15178 df-clim 15195 df-rlim 15196 df-sum 15396 df-ef 15775 df-sin 15777 df-cos 15778 df-pi 15780 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cncf 24039 df-limc 25028 df-dv 25029 df-log 25710 |
This theorem is referenced by: divlogrlim 25788 logno1 25789 logbleb 25931 logblt 25932 cxploglim 26125 cxploglim2 26126 emcllem4 26146 emcllem6 26148 chtge0 26259 isppw 26261 chtwordi 26303 fsumvma2 26360 chpval2 26364 chpchtsum 26365 chpub 26366 bposlem1 26430 chebbnd1lem1 26615 chebbnd1lem3 26617 chebbnd1 26618 chtppilimlem1 26619 chtppilimlem2 26620 chtppilim 26621 chebbnd2 26623 chto1lb 26624 rplogsumlem2 26631 rpvmasumlem 26633 vmalogdivsum2 26684 vmalogdivsum 26685 2vmadivsumlem 26686 chpdifbndlem1 26699 selberg3lem1 26703 selberg3 26705 selberg4lem1 26706 selberg4 26707 selberg3r 26715 selberg4r 26716 selberg34r 26717 pntrlog2bndlem1 26723 pntrlog2bndlem2 26724 pntrlog2bndlem3 26725 pntrlog2bndlem4 26726 pntrlog2bndlem5 26727 pntrlog2bndlem6 26729 pntrlog2bnd 26730 pntibndlem2 26737 pntlemb 26743 pntlemg 26744 pntlemh 26745 pntlemr 26748 pntlemj 26749 pntlemf 26751 pntlemo 26753 pnt 26760 ostth2lem3 26781 ostth2lem4 26782 ostth2 26783 ostth3 26784 hgt750leme 32634 aks4d1p1p7 40079 aks4d1p6 40086 |
Copyright terms: Public domain | W3C validator |