Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 12501 | . 2 ⊢ ℕ0 ⊆ ℂ | |
| 2 | nn0rei.1 | . 2 ⊢ 𝐴 ∈ ℕ0 | |
| 3 | 1, 2 | sselii 3975 | 1 ⊢ 𝐴 ∈ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2099 ℂcc 11130 ℕ0cn0 12496 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-mulcl 11194 ax-i2m1 11200 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-nn 12237 df-n0 12497 |
| This theorem is referenced by: nn0le2xi 12550 num0u 12712 num0h 12713 numsuc 12715 numsucc 12741 numma 12745 nummac 12746 numma2c 12747 numadd 12748 numaddc 12749 nummul1c 12750 nummul2c 12751 decrmanc 12758 decrmac 12759 decaddi 12761 decaddci 12762 decsubi 12764 decmul1 12765 decmulnc 12768 11multnc 12769 decmul10add 12770 6p5lem 12771 4t3lem 12798 7t3e21 12811 7t6e42 12814 8t3e24 12817 8t4e32 12818 8t8e64 12822 9t3e27 12824 9t4e36 12825 9t5e45 12826 9t6e54 12827 9t7e63 12828 9t11e99 12831 decbin0 12841 decbin2 12842 sq10 14249 3dec 14251 nn0le2msqi 14252 nn0opthlem1 14253 nn0opthi 14255 nn0opth2i 14256 faclbnd4lem1 14278 cats1fvn 14835 bpoly4 16029 fsumcube 16030 3dvdsdec 16302 3dvds2dec 16303 divalglem2 16365 3lcm2e6 16697 phiprmpw 16738 dec5dvds 17026 dec5dvds2 17027 dec2nprm 17029 modxai 17030 mod2xi 17031 mod2xnegi 17033 modsubi 17034 gcdi 17035 decexp2 17037 numexp0 17038 numexp1 17039 numexpp1 17040 numexp2x 17041 decsplit0b 17042 decsplit0 17043 decsplit1 17044 decsplit 17045 karatsuba 17046 2exp8 17051 prmlem2 17082 83prm 17085 139prm 17086 163prm 17087 631prm 17089 1259lem1 17093 1259lem2 17094 1259lem3 17095 1259lem4 17096 1259lem5 17097 1259prm 17098 2503lem1 17099 2503lem2 17100 2503lem3 17101 2503prm 17102 4001lem1 17103 4001lem2 17104 4001lem3 17105 4001lem4 17106 4001prm 17107 psdmul 22083 log2ublem1 26871 log2ublem2 26872 log2ublem3 26873 log2ub 26874 birthday 26879 ppidif 27088 bpos1lem 27208 9p10ne21 30273 dfdec100 32587 dp20u 32595 dp20h 32596 dpmul10 32612 dpmul100 32614 dp3mul10 32615 dpmul1000 32616 dpexpp1 32625 0dp2dp 32626 dpadd2 32627 dpadd 32628 dpmul 32630 dpmul4 32631 lmatfvlem 33410 ballotlemfp1 34105 ballotth 34151 reprlt 34245 hgt750lemd 34274 hgt750lem2 34278 subfacp1lem1 34783 poimirlem26 37113 poimirlem28 37115 420gcd8e4 41471 lcmeprodgcdi 41472 12lcm5e60 41473 60lcm7e420 41475 3exp7 41518 3lexlogpow5ineq1 41519 3lexlogpow5ineq5 41525 aks4d1p1p7 41539 aks4d1p1 41541 decaddcom 41852 sqn5i 41853 decpmulnc 41855 decpmul 41856 sqdeccom12 41857 sq3deccom12 41858 235t711 41861 ex-decpmul 41862 sq45 42089 sum9cubes 42090 resqrtvalex 43069 imsqrtvalex 43070 inductionexd 43579 unitadd 43619 fmtno5lem4 46890 257prm 46895 fmtno4prmfac 46906 fmtno5fac 46916 139prmALT 46930 127prm 46933 m11nprm 46935 11t31e341 47066 2exp340mod341 47067 ackval3012 47759 |
| Copyright terms: Public domain | W3C validator |