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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 11383 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ℂcc 10270 ℕcn 11374 |
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-1cn 10330 ax-addcl 10332 |
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-ral 3095 df-rex 3096 df-reu 3097 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-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 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-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 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-ov 6925 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-nn 11375 |
This theorem is referenced by: 9p1e10 11847 numnncl2 11869 dec10p 11889 3dec 13371 faclbnd4lem1 13398 4bc2eq6 13434 ef01bndlem 15316 3dvds 15459 divalglem8 15530 pockthi 16015 dec5nprm 16174 dec2nprm 16175 modxai 16176 modxp1i 16178 mod2xnegi 16179 modsubi 16180 23prm 16224 37prm 16226 43prm 16227 83prm 16228 139prm 16229 163prm 16230 1259lem1 16236 1259lem4 16239 2503lem2 16243 4001lem1 16246 4001lem3 16248 mcubic 25025 cubic2 25026 cubic 25027 quart1cl 25032 quart1lem 25033 quart1 25034 quartlem1 25035 quartlem2 25036 log2ublem1 25125 log2ublem2 25126 log2ub 25128 bclbnd 25457 bposlem8 25468 pntlemf 25746 ex-lcm 27890 dpmul10 30165 decdiv10 30166 dp3mul10 30168 dpadd2 30180 dpadd 30181 dpadd3 30182 dpmul 30183 dpmul4 30184 ballotlem2 31149 ballotlemfmpn 31155 ballotth 31198 cnndvlem1 33110 1t10e1p1e11 42352 deccarry 42353 fmtnoprmfac2lem1 42499 139prmALT 42532 3exp4mod41 42554 41prothprmlem1 42555 bgoldbtbndlem1 42718 tgblthelfgott 42728 tgoldbachlt 42729 |
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