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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 12153 | . 2 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
2 | gt0ne0 10840 | . 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 386 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4886 ℝcr 10271 0cc0 10272 < clt 10411 ℝ+crp 12137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-addrcl 10333 ax-rnegex 10343 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-rp 12138 |
This theorem is referenced by: rprene0 12156 rpcnne0 12157 rpne0d 12186 divge1 12207 xlemul1 12432 ltdifltdiv 12954 mulmod0 12995 negmod0 12996 moddiffl 13000 modid0 13015 modmuladd 13031 modmuladdnn0 13033 2txmodxeq0 13049 rpexpcl 13197 expnlbnd 13313 rennim 14386 sqrtdiv 14413 o1fsum 14949 divrcnv 14988 rpmsubg 20206 itg2const2 23945 reeff1o 24638 reefgim 24641 logne0 24763 advlog 24837 advlogexp 24838 logcxp 24852 cxprec 24869 cxpmul 24871 abscxp 24875 cxple2 24880 dvcxp1 24921 dvcxp2 24922 dvsqrt 24923 relogbreexp 24953 relogbzexp 24954 relogbmul 24955 relogbdiv 24957 relogbexp 24958 relogbcxp 24963 relogbcxpb 24965 relogbf 24969 logblog 24970 logbgt0b 24971 rlimcnp 25144 efrlim 25148 cxplim 25150 cxp2limlem 25154 cxploglim 25156 logdifbnd 25172 logdiflbnd 25173 logfacrlim2 25403 bposlem8 25468 vmadivsum 25623 mudivsum 25671 mulogsumlem 25672 logdivsum 25674 log2sumbnd 25685 selberg2lem 25691 selberg2 25692 pntrmax 25705 selbergr 25709 pntrlog2bndlem4 25721 pntrlog2bndlem5 25722 pntlem3 25750 padicabvcxp 25773 blocnilem 28231 nmcexi 29457 probfinmeasbOLD 31089 probfinmeasb 31090 signsplypnf 31227 logdivsqrle 31330 poimirlem29 34064 areacirclem1 34125 areacirclem4 34128 areacirc 34130 heiborlem6 34239 heiborlem7 34240 xralrple2 40478 recnnltrp 40501 rpgtrecnn 40505 ioodvbdlimc1lem2 41075 ioodvbdlimc2lem 41077 fldivmod 43328 relogbmulbexp 43370 relogbdivb 43371 blenre 43383 |
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