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Mirrors > Home > ILE Home > Th. List > nn0cn | GIF version |
Description: A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nn0cn | ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0sscn 8940 | . 2 ⊢ ℕ0 ⊆ ℂ | |
2 | 1 | sseli 3063 | 1 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ℂcc 7586 ℕ0cn0 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-rnegex 7697 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-int 3742 df-inn 8685 df-n0 8936 |
This theorem is referenced by: nn0nnaddcl 8966 elnn0nn 8977 nn0n0n1ge2 9079 uzaddcl 9337 fzctr 9865 nn0split 9868 zpnn0elfzo1 9940 ubmelm1fzo 9958 subfzo0 9974 modqmuladdnn0 10096 addmodidr 10101 modfzo0difsn 10123 nn0ennn 10161 expadd 10290 expmul 10293 bernneq 10367 bernneq2 10368 faclbnd 10442 faclbnd6 10445 bccmpl 10455 bcn0 10456 bcnn 10458 bcnp1n 10460 bcn2 10465 bcp1m1 10466 bcpasc 10467 bcn2p1 10471 hashfzo0 10524 hashfz0 10526 fisum0diag2 11171 hashiun 11202 binom1dif 11211 bcxmas 11213 geolim 11235 efaddlem 11294 efexp 11302 eftlub 11310 demoivreALT 11394 nn0ob 11517 modremain 11538 mulgcdr 11618 nn0seqcvgd 11634 znnen 11822 ennnfonelemp1 11830 |
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