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Mirrors > Home > MPE Home > Th. List > rprege0 | Structured version Visualization version GIF version |
Description: A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
rprege0 | ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 13030 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rpge0 13035 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) | |
3 | 1, 2 | jca 510 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 class class class wbr 5145 ℝcr 11148 0cc0 11149 ≤ cle 11290 ℝ+crp 13022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-addrcl 11210 ax-rnegex 11220 ax-cnre 11222 ax-pre-lttri 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-rp 13023 |
This theorem is referenced by: resqrex 15250 sqrtdiv 15265 o1fsum 15812 prmreclem3 16915 aaliou3lem3 26369 pige3ALT 26544 rpcxpcl 26700 cxprec 26710 harmoniclbnd 27034 harmonicbnd4 27036 basellem4 27109 logfaclbnd 27248 logfacrlim 27250 logexprlim 27251 bposlem7 27316 vmadivsum 27508 dchrisum0lem2a 27543 dchrisum0lem2 27544 dchrisum0 27546 mudivsum 27556 mulogsumlem 27557 selberglem2 27572 selberg2lem 27576 pntrsumo1 27591 minvecolem3 30806 ehl2eudis0lt 48150 itsclc0 48195 itsclc0b 48196 |
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