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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 11724 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ℂcc 10613 ℕcn 11716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 ax-1cn 10673 ax-addcl 10675 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-nn 11717 |
This theorem is referenced by: 9p1e10 12181 numnncl2 12202 dec10p 12222 3dec 13718 faclbnd4lem1 13745 4bc2eq6 13781 ef01bndlem 15629 3dvds 15776 divalglem8 15845 pockthi 16343 dec5nprm 16502 dec2nprm 16503 modxai 16504 modxp1i 16506 mod2xnegi 16507 modsubi 16508 23prm 16555 37prm 16557 43prm 16558 83prm 16559 139prm 16560 163prm 16561 1259lem1 16567 1259lem4 16570 2503lem2 16574 4001lem1 16577 4001lem3 16579 mcubic 25585 cubic2 25586 cubic 25587 quart1cl 25592 quart1lem 25593 quart1 25594 quartlem1 25595 quartlem2 25596 log2ublem1 25684 log2ublem2 25685 log2ub 25687 bclbnd 26016 bposlem8 26027 pntlemf 26341 ex-lcm 28395 dpmul10 30744 decdiv10 30745 dp3mul10 30747 dpadd2 30759 dpadd 30760 dpadd3 30761 dpmul 30762 dpmul4 30763 ballotlem2 32025 ballotlemfmpn 32031 ballotth 32074 cnndvlem1 34355 addassnni 39613 addcomnni 39614 mulassnni 39615 mulcomnni 39616 gcdaddmzz2nncomi 39624 lcmeprodgcdi 39635 lcmineqlem6 39662 lcmineqlem23 39679 3lexlogpow5ineq5 39688 1t10e1p1e11 44336 deccarry 44337 fmtnoprmfac2lem1 44552 139prmALT 44582 3exp4mod41 44602 41prothprmlem1 44603 2exp340mod341 44719 bgoldbtbndlem1 44791 tgblthelfgott 44801 tgoldbachlt 44802 |
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