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Mirrors > Home > MPE Home > Th. List > rpcnne0 | Structured version Visualization version GIF version |
Description: A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.) |
Ref | Expression |
---|---|
rpcnne0 | ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpcn 12917 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
2 | rpne0 12923 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0) | |
3 | 1, 2 | jca 512 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ≠ wne 2941 ℂcc 11045 0cc0 11047 ℝ+crp 12907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-resscn 11104 ax-1cn 11105 ax-addrcl 11108 ax-rnegex 11118 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-ltxr 11190 df-rp 12908 |
This theorem is referenced by: rpcndif0 12926 mod0 13773 modlt 13777 modcyc 13803 modmuladdnn0 13812 moddi 13836 modirr 13839 aaliou3lem3 25688 aaliou3lem8 25689 reeff1o 25790 reeflog 25920 relogeftb 25924 rpcxpcl 26015 relogbcxp 26119 rlimcnp 26299 rlimcnp2 26300 divsqrtsumlem 26313 harmonicbnd4 26344 logfacrlim 26556 logexprlim 26557 vmadivsum 26814 dchrmusum2 26826 dchrvmasumlem2 26830 dchrvmasumiflem1 26833 dchrisum0lem2a 26849 mudivsum 26862 mulogsumlem 26863 mulog2sumlem2 26867 selberglem2 26878 selberg2lem 26882 selberg2 26883 pntrsumo1 26897 selbergr 26900 pntibndlem2 26923 pntibndlem3 26924 pntlemb 26929 pntlemr 26934 pntlemf 26937 blocnilem 29632 minvecolem3 29704 itg2addnclem2 36097 fllogbd 46578 |
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