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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.) |
Ref | Expression |
---|---|
nn0rei.1 | ⊢ 𝐴 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0cni | ⊢ 𝐴 ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0rei.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
2 | 1 | nn0rei 11515 | . 2 ⊢ 𝐴 ∈ ℝ |
3 | 2 | recni 10264 | 1 ⊢ 𝐴 ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2139 ℂcc 10146 ℕ0cn0 11504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-i2m1 10216 ax-1ne0 10217 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-nn 11233 df-n0 11505 |
This theorem is referenced by: nn0le2xi 11559 num0u 11720 num0h 11721 numsuc 11723 numsucc 11761 numma 11769 nummac 11770 numma2c 11771 numadd 11772 numaddc 11773 nummul1c 11774 nummul2c 11775 decrmanc 11788 decrmac 11789 decaddi 11791 decaddci 11792 decsubi 11795 decsubiOLD 11796 decmul1 11797 decmul1OLD 11798 decmulnc 11803 11multnc 11804 decmul10add 11805 decmul10addOLD 11806 6p5lem 11807 4t3lem 11843 6t5e30OLD 11857 7t3e21 11861 7t6e42 11864 8t3e24 11867 8t4e32 11868 8t8e64 11874 9t3e27 11876 9t4e36 11877 9t5e45 11878 9t6e54 11879 9t7e63 11880 9t11e99 11883 decbin0 11894 decbin2 11895 sq10 13262 3dec 13264 3decOLD 13267 nn0le2msqi 13268 nn0opthlem1 13269 nn0opthi 13271 nn0opth2i 13272 faclbnd4lem1 13294 cats1fvn 13823 bpoly4 15009 fsumcube 15010 3dvdsdec 15276 3dvdsdecOLD 15277 3dvds2dec 15278 3dvds2decOLD 15279 divalglem2 15340 3lcm2e6 15662 phiprmpw 15703 dec5dvds 15990 dec5dvds2 15991 dec2nprm 15993 modxai 15994 mod2xi 15995 mod2xnegi 15997 modsubi 15998 gcdi 15999 decexp2 16001 numexp0 16002 numexp1 16003 numexpp1 16004 numexp2x 16005 decsplit0b 16006 decsplit0 16007 decsplit1 16008 decsplit 16009 decsplit0bOLD 16010 decsplit0OLD 16011 decsplit1OLD 16012 decsplitOLD 16013 karatsuba 16014 karatsubaOLD 16015 2exp8 16018 prmlem2 16049 83prm 16052 139prm 16053 163prm 16054 631prm 16056 1259lem1 16060 1259lem2 16061 1259lem3 16062 1259lem4 16063 1259lem5 16064 1259prm 16065 2503lem1 16066 2503lem2 16067 2503lem3 16068 2503prm 16069 4001lem1 16070 4001lem2 16071 4001lem3 16072 4001lem4 16073 4001prm 16074 log2ublem1 24893 log2ublem2 24894 log2ublem3 24895 log2ub 24896 birthday 24901 ppidif 25109 bpos1lem 25227 dfdec100 29906 dp20u 29915 dp20h 29916 dpmul10 29933 dpmul100 29935 dp3mul10 29936 dpmul1000 29937 dpexpp1 29946 0dp2dp 29947 dpadd2 29948 dpadd 29949 dpmul 29951 dpmul4 29952 lmatfvlem 30211 ballotlemfp1 30883 ballotth 30929 reprlt 31027 hgt750lemd 31056 hgt750lem2 31060 subfacp1lem1 31489 poimirlem26 33766 poimirlem28 33768 inductionexd 38973 unitadd 39018 fmtno5lem4 41996 257prm 42001 fmtno4prmfac 42012 fmtno5fac 42022 139prmALT 42039 127prm 42043 m11nprm 42046 |
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