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Mirrors > Home > MPE Home > Th. List > rpexpcl | Structured version Visualization version Unicode version |
Description: Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.) |
Ref | Expression |
---|---|
rpexpcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . 2 | |
2 | rpne0 11848 | . . 3 | |
3 | 2 | adantr 481 | . 2 |
4 | simpr 477 | . 2 | |
5 | rpssre 11843 | . . . 4 | |
6 | ax-resscn 9993 | . . . 4 | |
7 | 5, 6 | sstri 3612 | . . 3 |
8 | rpmulcl 11855 | . . 3 | |
9 | 1rp 11836 | . . 3 | |
10 | rpreccl 11857 | . . . 4 | |
11 | 10 | adantr 481 | . . 3 |
12 | 7, 8, 9, 11 | expcl2lem 12872 | . 2 |
13 | 1, 3, 4, 12 | syl3anc 1326 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 wne 2794 (class class class)co 6650 cc 9934 cr 9935 cc0 9936 c1 9937 cdiv 10684 cz 11377 crp 11832 cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 |
This theorem is referenced by: expgt0 12893 ltexp2a 12912 expcan 12913 ltexp2 12914 leexp2a 12916 ltexp2r 12917 expnlbnd2 12995 rpexpcld 13032 expcnv 14596 effsumlt 14841 ef01bndlem 14914 rpnnen2lem11 14953 iscmet3lem3 23088 iscmet3lem1 23089 iscmet3lem2 23090 iscmet3 23091 minveclem3 23200 pjthlem1 23208 aaliou3lem1 24097 aaliou3lem2 24098 aaliou3lem3 24099 aaliou3lem8 24100 aaliou3lem5 24102 aaliou3lem6 24103 aaliou3lem7 24104 aaliou3lem9 24105 tanregt0 24285 asinlem3 24598 cxp2limlem 24702 ftalem5 24803 basellem3 24809 basellem4 24810 basellem8 24814 chebbnd1lem3 25160 dchrisum0lem1a 25175 dchrisum0lem1b 25204 dchrisum0lem1 25205 dchrisum0lem2a 25206 dchrisum0lem2 25207 dchrisum0lem3 25208 pntlemd 25283 pntlema 25285 pntlemb 25286 pntlemh 25288 pntlemr 25291 pntlemi 25293 pntlemf 25294 pntlemo 25296 pntlem3 25298 pntleml 25300 ostth2lem1 25307 ostth3 25327 minvecolem3 27732 pjhthlem1 28250 dpexpp1 29616 dya2icoseg 30339 faclimlem3 31631 geomcau 33555 dignnld 42397 |
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