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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 6925 | . 2 ⊢ (𝐴 ∈ ℝ+ → ((log ↾ ℝ+)‘𝐴) = (log‘𝐴)) | |
| 2 | relogf1o 26608 | . . . 4 ⊢ (log ↾ ℝ+):ℝ+–1-1-onto→ℝ | |
| 3 | f1of 6848 | . . . 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 | ffvelcdmi 7103 | . 2 ⊢ (𝐴 ∈ ℝ+ → ((log ↾ ℝ+)‘𝐴) ∈ ℝ) |
| 6 | 1, 5 | eqeltrrd 2842 | 1 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ↾ cres 5687 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 ℝcr 11154 ℝ+crp 13034 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-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: logneg 26630 lognegb 26632 relogoprlem 26633 reexplog 26637 relogexp 26638 logfac 26643 logleb 26645 rplogcl 26646 logmul2 26658 logdiv2 26659 abslogle 26660 logdivlti 26662 logdivlt 26663 logdivle 26664 relogcld 26665 advlog 26696 advlogexp 26697 logccv 26705 logcxp 26711 rpcxpcl 26718 cxpmul 26730 abscxp 26734 cxple2 26739 logsqrt 26746 dvcxp1 26782 dvcxp2 26783 loglesqrt 26804 relogbcl 26816 relogbmul 26820 logbgt0b 26836 log2ub 26992 log2le1 26993 birthday 26997 cxploglim 27021 cxploglim2 27022 amgmlem 27033 logdifbnd 27037 emcllem7 27045 emre 27049 emgt0 27050 harmonicbnd3 27051 harmoniclbnd 27052 harmonicbnd4 27054 relgamcl 27105 cht2 27215 chtleppi 27254 chtublem 27255 chtub 27256 logfacubnd 27265 logfaclbnd 27266 logfacbnd3 27267 logfacrlim 27268 logexprlim 27269 efexple 27325 bposlem6 27333 bposlem7 27334 bposlem8 27335 bposlem9 27336 chebbnd1lem3 27515 chebbnd1 27516 chto1ub 27520 vmadivsum 27526 rpvmasumlem 27531 dchrvmasumlem2 27542 dchrvmasumlema 27544 dchrvmasumiflem1 27545 dchrvmasumiflem2 27546 dchrisum0fno1 27555 rpvmasum2 27556 dchrisum0re 27557 rpvmasum 27570 rplogsum 27571 dirith2 27572 logdivsum 27577 mulog2sumlem2 27579 mulog2sumlem3 27580 logsqvma 27586 log2sumbnd 27588 selberglem1 27589 selberglem2 27590 selberglem3 27591 selberg 27592 selberg2lem 27594 selberg2 27595 pntrsumo1 27609 selbergr 27612 pntrlog2bndlem4 27624 pntibndlem3 27636 xrge0iifiso 33934 logdivsqrle 34665 hgt750lem 34666 hgt750lemb 34671 reglogcl 42901 reglogltb 42902 reglogleb 42903 reglogmul 42904 reglogexp 42905 reglogbas 42906 reglog1 42907 stirlinglem12 46100 stirlinglem13 46101 stirlinglem14 46102 lighneallem2 47593 logbge0b 48484 logblt1b 48485 |
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