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Mirrors > Home > MPE Home > Th. List > expcl | Structured version Visualization version Unicode version |
Description: Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.) |
Ref | Expression |
---|---|
expcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3624 | . 2 | |
2 | mulcl 10020 | . 2 | |
3 | ax-1cn 9994 | . 2 | |
4 | 1, 2, 3 | expcllem 12871 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 (class class class)co 6650 cc 9934 cn0 11292 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-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-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: expeq0 12890 expnegz 12894 mulexp 12899 mulexpz 12900 expadd 12902 expaddzlem 12903 expaddz 12904 expmul 12905 expmulz 12906 expdiv 12911 binom3 12985 digit2 12997 digit1 12998 expcld 13008 faclbnd2 13078 faclbnd4lem4 13083 faclbnd6 13086 cjexp 13890 absexp 14044 ackbijnn 14560 binomlem 14561 binom1p 14563 binom1dif 14565 expcnv 14596 geolim 14601 geolim2 14602 geo2sum 14604 geomulcvg 14607 geoisum 14608 geoisumr 14609 geoisum1 14610 geoisum1c 14611 0.999... 14612 0.999...OLD 14613 fallrisefac 14756 0risefac 14769 binomrisefac 14773 bpolysum 14784 bpolydiflem 14785 fsumkthpow 14787 bpoly3 14789 bpoly4 14790 fsumcube 14791 eftcl 14804 eftabs 14806 efcllem 14808 efcj 14822 efaddlem 14823 eflegeo 14851 efi4p 14867 prmreclem6 15625 decsplitOLD 15791 karatsuba 15792 karatsubaOLD 15793 expmhm 19815 mbfi1fseqlem6 23487 itg0 23546 itgz 23547 itgcl 23550 itgcnlem 23556 itgsplit 23602 dvexp 23716 dvexp3 23741 plyf 23954 ply1termlem 23959 plypow 23961 plyeq0lem 23966 plypf1 23968 plyaddlem1 23969 plymullem1 23970 coeeulem 23980 coeidlem 23993 coeid3 23996 plyco 23997 dgrcolem2 24030 plycjlem 24032 plyrecj 24035 vieta1 24067 elqaalem3 24076 aareccl 24081 aalioulem1 24087 geolim3 24094 psergf 24166 dvradcnv 24175 psercn2 24177 pserdvlem2 24182 pserdv2 24184 abelthlem4 24188 abelthlem5 24189 abelthlem6 24190 abelthlem7 24192 abelthlem9 24194 advlogexp 24401 logtayllem 24405 logtayl 24406 logtaylsum 24407 logtayl2 24408 cxpeq 24498 dcubic1lem 24570 dcubic2 24571 dcubic1 24572 dcubic 24573 mcubic 24574 cubic2 24575 cubic 24576 binom4 24577 dquartlem2 24579 dquart 24580 quart1cl 24581 quart1lem 24582 quart1 24583 quartlem1 24584 quartlem2 24585 quart 24588 atantayl 24664 atantayl2 24665 atantayl3 24666 leibpi 24669 log2cnv 24671 log2tlbnd 24672 log2ublem3 24675 ftalem1 24799 ftalem4 24802 ftalem5 24803 basellem3 24809 musum 24917 1sgmprm 24924 perfect 24956 lgsquadlem1 25105 rplogsumlem2 25174 ostth2lem2 25323 numclwlk3lem3 27206 ipval2 27562 dipcl 27567 dipcn 27575 subfacval2 31169 jm2.23 37563 lhe4.4ex1a 38528 perfectALTV 41632 altgsumbc 42130 altgsumbcALT 42131 nn0digval 42394 |
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