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Mirrors > Home > MPE Home > Th. List > rprecred | Structured version Visualization version Unicode version |
Description: Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 |
Ref | Expression |
---|---|
rprecred |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . . 3 | |
2 | 1 | rpreccld 11882 | . 2 |
3 | 2 | rpred 11872 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1990 (class class class)co 6650 cr 9935 c1 9937 cdiv 10684 crp 11832 |
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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-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-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-rp 11833 |
This theorem is referenced by: xov1plusxeqvd 12318 ltexp2r 12917 expnlbnd2 12995 rlimno1 14384 lebnumii 22765 sca2rab 23280 aalioulem4 24090 aalioulem5 24091 dvradcnv 24175 tanregt0 24285 divlogrlim 24381 logccv 24409 cxplt3 24446 asinlem3 24598 rlimcxp 24700 cxp2lim 24703 divsqrtsumlem 24706 logdiflbnd 24721 lgamgulmlem2 24756 lgamgulmlem3 24757 basellem3 24809 dchrisum0lema 25203 dchrisum0lem1 25205 dchrisum0lem2a 25206 mulog2sumlem1 25223 vmalogdivsum2 25227 pntrlog2bndlem2 25267 pntlemd 25283 pntlemr 25291 ostth3 25327 nmcexi 28885 knoppndvlem18 32520 knoppndvlem20 32522 irrapxlem4 37389 irrapxlem5 37390 ioodvbdlimc1lem2 40147 ioodvbdlimc2lem 40149 stoweidlem14 40231 fourierdlem39 40363 pimrecltpos 40919 smfrec 40996 |
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