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Mirrors > Home > ILE Home > Th. List > nn0cnd | GIF version |
Description: A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nn0red.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0cnd | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0red.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
2 | 1 | nn0red 9261 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | recnd 8017 | 1 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ℂcc 7840 ℕ0cn0 9207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-sep 4136 ax-cnex 7933 ax-resscn 7934 ax-1re 7936 ax-addrcl 7939 ax-rnegex 7951 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-int 3860 df-inn 8951 df-n0 9208 |
This theorem is referenced by: modsumfzodifsn 10429 addmodlteq 10431 uzennn 10469 expaddzaplem 10597 expaddzap 10598 expmulzap 10600 nn0le2msqd 10734 nn0opthlem1d 10735 nn0opthd 10737 nn0opth2d 10738 facdiv 10753 bcp1n 10776 bcn2m1 10784 bcn2p1 10785 omgadd 10817 fihashssdif 10833 hashdifpr 10835 hashxp 10841 zfz1isolemsplit 10853 zfz1isolem1 10855 fsumconst 11497 hash2iun1dif1 11523 binomlem 11526 bcxmas 11532 arisum 11541 arisum2 11542 mertensabs 11580 effsumlt 11735 dvdsexp 11902 nn0ob 11948 divalglemnn 11958 divalgmod 11967 bezoutlemnewy 12032 bezoutlema 12035 bezoutlemb 12036 mulgcd 12052 absmulgcd 12053 mulgcdr 12054 gcddiv 12055 lcmgcd 12113 lcmid 12115 lcm1 12116 3lcm2e6woprm 12121 6lcm4e12 12122 mulgcddvds 12129 qredeu 12132 divgcdcoprm0 12136 divgcdcoprmex 12137 cncongr1 12138 cncongr2 12139 pw2dvdseulemle 12202 phiprmpw 12257 eulerthlema 12265 prmdiveq 12271 odzdvds 12280 powm2modprm 12287 coprimeprodsq 12292 pceulem 12329 pczpre 12332 pcqmul 12338 pcaddlem 12374 pcmpt 12378 pcmpt2 12379 sumhashdc 12382 pcfac 12385 oddprmdvds 12389 mul4sq 12429 4sqlem12 12437 mulgnn0dir 13109 mulgnn0ass 13115 lgslem1 14879 lgsvalmod 14898 lgseisenlem2 14929 m1lgs 14930 2sqlem8 14948 |
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