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Mirrors > Home > ILE Home > Th. List > relogcl | GIF version |
Description: Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
Ref | Expression |
---|---|
relogcl | β’ (π΄ β β+ β (logβπ΄) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 5541 | . 2 β’ (π΄ β β+ β ((log βΎ β+)βπ΄) = (logβπ΄)) | |
2 | relogf1o 14422 | . . . 4 β’ (log βΎ β+):β+β1-1-ontoββ | |
3 | f1of 5463 | . . . 4 β’ ((log βΎ β+):β+β1-1-ontoββ β (log βΎ β+):β+βΆβ) | |
4 | 2, 3 | ax-mp 5 | . . 3 β’ (log βΎ β+):β+βΆβ |
5 | 4 | ffvelcdmi 5653 | . 2 β’ (π΄ β β+ β ((log βΎ β+)βπ΄) β β) |
6 | 1, 5 | eqeltrrd 2255 | 1 β’ (π΄ β β+ β (logβπ΄) β β) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wcel 2148 βΎ cres 4630 βΆwf 5214 β1-1-ontoβwf1o 5217 βcfv 5218 βcr 7813 β+crp 9656 logclog 14417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 ax-caucvg 7934 ax-pre-suploc 7935 ax-addf 7936 ax-mulf 7937 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-disj 3983 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-of 6086 df-1st 6144 df-2nd 6145 df-recs 6309 df-irdg 6374 df-frec 6395 df-1o 6420 df-oadd 6424 df-er 6538 df-map 6653 df-pm 6654 df-en 6744 df-dom 6745 df-fin 6746 df-sup 6986 df-inf 6987 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-q 9623 df-rp 9657 df-xneg 9775 df-xadd 9776 df-ioo 9895 df-ico 9897 df-icc 9898 df-fz 10012 df-fzo 10146 df-seqfrec 10449 df-exp 10523 df-fac 10709 df-bc 10731 df-ihash 10759 df-shft 10827 df-cj 10854 df-re 10855 df-im 10856 df-rsqrt 11010 df-abs 11011 df-clim 11290 df-sumdc 11365 df-ef 11659 df-e 11660 df-rest 12696 df-topgen 12715 df-psmet 13587 df-xmet 13588 df-met 13589 df-bl 13590 df-mopn 13591 df-top 13638 df-topon 13651 df-bases 13683 df-ntr 13736 df-cn 13828 df-cnp 13829 df-tx 13893 df-cncf 14198 df-limced 14265 df-dvap 14266 df-relog 14419 |
This theorem is referenced by: reeflog 14424 relogef 14425 relogoprlem 14429 reexplog 14432 relogexp 14433 logleb 14436 logrpap0b 14437 rplogcl 14440 logdivlti 14442 relogcld 14443 logcxp 14458 rpcncxpcl 14463 rpcxpcl 14464 cxpap0 14465 rpcxpadd 14466 cxpmul 14473 abscxp 14475 logsqrt 14483 rplogbid1 14505 rpelogb 14507 rprelogbmul 14513 |
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