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Mirrors > Home > MPE Home > Th. List > rpne0 | Structured version Visualization version GIF version |
Description: A positive real is nonzero. (Contributed by NM, 18-Jul-2008.) |
Ref | Expression |
---|---|
rpne0 | ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpregt0 13028 | . 2 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
2 | gt0ne0 11716 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ℝcr 11144 0cc0 11145 < clt 11285 ℝ+crp 13014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11202 ax-1cn 11203 ax-addrcl 11206 ax-rnegex 11216 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-ltxr 11290 df-rp 13015 |
This theorem is referenced by: rprene0 13031 rpcnne0 13032 rpne0d 13061 divge1 13082 xlemul1 13309 ltdifltdiv 13840 mulmod0 13883 negmod0 13884 moddiffl 13888 modid0 13903 modmuladd 13919 modmuladdnn0 13921 2txmodxeq0 13937 rpexpcl 14086 expnlbnd 14236 rennim 15230 sqrtdiv 15256 o1fsum 15803 divrcnv 15842 rpmsubg 21398 itg2const2 25732 reeff1o 26446 logne0 26575 advlog 26650 advlogexp 26651 logcxp 26665 cxprec 26682 cxpmul 26684 abscxp 26688 cxple2 26693 dvcxp1 26736 dvcxp2 26737 dvsqrt 26738 relogbreexp 26772 relogbzexp 26773 relogbmul 26774 relogbdiv 26776 relogbexp 26777 relogbcxp 26782 relogbcxpb 26784 relogbf 26788 logbgt0b 26790 rlimcnp 26962 efrlim 26966 efrlimOLD 26967 cxplim 26969 cxp2limlem 26973 cxploglim 26975 logdifbnd 26991 logdiflbnd 26992 logfacrlim2 27224 bposlem8 27289 vmadivsum 27480 mudivsum 27528 mulogsumlem 27529 logdivsum 27531 log2sumbnd 27542 selberg2lem 27548 selberg2 27549 pntrmax 27562 selbergr 27566 pntrlog2bndlem4 27578 pntrlog2bndlem5 27579 pntlem3 27607 padicabvcxp 27630 blocnilem 30706 nmcexi 31928 probfinmeasb 34199 probfinmeasbALTV 34200 signsplypnf 34333 logdivsqrle 34433 poimirlem29 37273 areacirclem1 37332 areacirclem4 37335 areacirc 37337 heiborlem6 37440 heiborlem7 37441 dvrelog2 41687 dvrelog3 41688 aks4d1p1p6 41696 xralrple2 44879 recnnltrp 44902 rpgtrecnn 44905 ioodvbdlimc1lem2 45463 ioodvbdlimc2lem 45465 fldivmod 47782 relogbmulbexp 47825 relogbdivb 47826 blenre 47838 |
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