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| Mirrors > Home > MPE Home > Th. List > nncni | Structured version Visualization version GIF version | ||
| Description: A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
| Ref | Expression |
|---|---|
| nnre.1 | ⊢ 𝐴 ∈ ℕ |
| Ref | Expression |
|---|---|
| nncni | ⊢ 𝐴 ∈ ℂ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nnre.1 | . 2 ⊢ 𝐴 ∈ ℕ | |
| 2 | nncn 12274 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ℂcc 11153 ℕcn 12266 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 |
| This theorem is referenced by: 9p1e10 12735 numnncl2 12756 dec10p 12776 3dec 14305 faclbnd4lem1 14332 4bc2eq6 14368 ef01bndlem 16220 3dvds 16368 divalglem8 16437 pockthi 16945 dec5nprm 17104 dec2nprm 17105 modxai 17106 modxp1i 17108 mod2xnegi 17109 modsubi 17110 23prm 17156 37prm 17158 43prm 17159 83prm 17160 139prm 17161 163prm 17162 1259lem1 17168 1259lem4 17171 2503lem2 17175 4001lem1 17178 4001lem3 17180 mcubic 26890 cubic2 26891 cubic 26892 quart1cl 26897 quart1lem 26898 quart1 26899 quartlem1 26900 quartlem2 26901 log2ublem1 26989 log2ublem2 26990 log2ub 26992 bclbnd 27324 bposlem8 27335 pntlemf 27649 ex-lcm 30477 dpmul10 32877 decdiv10 32878 dp3mul10 32880 dpadd2 32892 dpadd 32893 dpadd3 32894 dpmul 32895 dpmul4 32896 ballotlem2 34491 ballotlemfmpn 34497 ballotth 34540 cnndvlem1 36538 addassnni 41985 addcomnni 41986 mulassnni 41987 mulcomnni 41988 gcdaddmzz2nncomi 41996 lcmeprodgcdi 42008 lcmineqlem6 42035 lcmineqlem23 42052 3lexlogpow5ineq5 42061 1t10e1p1e11 47322 deccarry 47323 fmtnoprmfac2lem1 47553 139prmALT 47583 3exp4mod41 47603 41prothprmlem1 47604 2exp340mod341 47720 bgoldbtbndlem1 47792 tgblthelfgott 47802 tgoldbachlt 47803 |
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