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| Mirrors > Home > MPE Home > Th. List > nn0cni | Structured version Visualization version GIF version | ||
| Description: A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022.) |
| Ref | Expression |
|---|---|
| nn0rei.1 | ⊢ 𝐴 ∈ ℕ0 |
| Ref | Expression |
|---|---|
| nn0cni | ⊢ 𝐴 ∈ ℂ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nn0sscn 12531 | . 2 ⊢ ℕ0 ⊆ ℂ | |
| 2 | nn0rei.1 | . 2 ⊢ 𝐴 ∈ ℕ0 | |
| 3 | 1, 2 | sselii 3980 | 1 ⊢ 𝐴 ∈ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ℂcc 11153 ℕ0cn0 12526 |
| 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-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| 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 df-n0 12527 |
| This theorem is referenced by: num0u 12744 num0h 12745 numsuc 12747 numsucc 12773 numma 12777 nummac 12778 numma2c 12779 numadd 12780 numaddc 12781 nummul1c 12782 nummul2c 12783 decrmanc 12790 decrmac 12791 decaddi 12793 decaddci 12794 decsubi 12796 decmul1 12797 decmulnc 12800 11multnc 12801 decmul10add 12802 6p5lem 12803 4t3lem 12830 7t3e21 12843 7t6e42 12846 8t3e24 12849 8t4e32 12850 8t8e64 12854 9t3e27 12856 9t4e36 12857 9t5e45 12858 9t6e54 12859 9t7e63 12860 9t11e99 12863 decbin0 12873 decbin2 12874 sq10 14303 3dec 14305 nn0le2msqi 14306 nn0opthlem1 14307 nn0opthi 14309 nn0opth2i 14310 faclbnd4lem1 14332 cats1fvn 14897 bpoly4 16095 fsumcube 16096 3dvdsdec 16369 3dvds2dec 16370 divalglem2 16432 3lcm2e6 16769 phiprmpw 16813 dec5dvds 17102 dec5dvds2 17103 dec2nprm 17105 modxai 17106 mod2xi 17107 mod2xnegi 17109 modsubi 17110 gcdi 17111 numexp0 17113 numexp1 17114 numexpp1 17115 numexp2x 17116 decsplit0b 17117 decsplit0 17118 decsplit1 17119 decsplit 17120 karatsuba 17121 2exp8 17126 prmlem2 17157 83prm 17160 139prm 17161 163prm 17162 631prm 17164 1259lem1 17168 1259lem2 17169 1259lem3 17170 1259lem4 17171 1259lem5 17172 1259prm 17173 2503lem1 17174 2503lem2 17175 2503lem3 17176 2503prm 17177 4001lem1 17178 4001lem2 17179 4001lem3 17180 4001lem4 17181 4001prm 17182 psdmul 22170 log2ublem1 26989 log2ublem2 26990 log2ublem3 26991 log2ub 26992 birthday 26997 ppidif 27206 bpos1lem 27326 9p10ne21 30489 dfdec100 32832 dp20u 32860 dp20h 32861 dpmul10 32877 dpmul100 32879 dp3mul10 32880 dpmul1000 32881 dpexpp1 32890 0dp2dp 32891 dpadd2 32892 dpadd 32893 dpmul 32895 dpmul4 32896 lmatfvlem 33814 ballotlemfp1 34494 ballotth 34540 reprlt 34634 hgt750lemd 34663 hgt750lem2 34667 subfacp1lem1 35184 poimirlem26 37653 poimirlem28 37655 420gcd8e4 42007 lcmeprodgcdi 42008 12lcm5e60 42009 60lcm7e420 42011 3exp7 42054 3lexlogpow5ineq1 42055 3lexlogpow5ineq5 42061 aks4d1p1p7 42075 aks4d1p1 42077 decaddcom 42319 sqn5i 42320 decpmulnc 42322 decpmul 42323 sqdeccom12 42324 sq3deccom12 42325 235t711 42339 ex-decpmul 42340 sq45 42681 sum9cubes 42682 resqrtvalex 43658 imsqrtvalex 43659 inductionexd 44168 unitadd 44208 fmtno5lem4 47543 257prm 47548 fmtno4prmfac 47559 fmtno5fac 47569 139prmALT 47583 127prm 47586 m11nprm 47588 11t31e341 47719 2exp340mod341 47720 ackval3012 48613 |
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