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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 8935 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | recnd 7718 | 1 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1463 ℂcc 7545 ℕ0cn0 8881 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-cnex 7636 ax-resscn 7637 ax-1re 7639 ax-addrcl 7642 ax-rnegex 7654 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-sn 3499 df-int 3738 df-inn 8631 df-n0 8882 |
This theorem is referenced by: modsumfzodifsn 10062 addmodlteq 10064 uzennn 10102 expaddzaplem 10229 expaddzap 10230 expmulzap 10232 nn0le2msqd 10358 nn0opthlem1d 10359 nn0opthd 10361 nn0opth2d 10362 facdiv 10377 bcp1n 10400 bcn2m1 10408 bcn2p1 10409 omgadd 10441 fihashssdif 10457 hashdifpr 10459 hashxp 10465 zfz1isolemsplit 10474 zfz1isolem1 10476 fsumconst 11115 hash2iun1dif1 11141 binomlem 11144 bcxmas 11150 arisum 11159 arisum2 11160 mertensabs 11198 effsumlt 11249 dvdsexp 11407 nn0ob 11453 divalglemnn 11463 divalgmod 11472 bezoutlemnewy 11530 bezoutlema 11533 bezoutlemb 11534 mulgcd 11550 absmulgcd 11551 mulgcdr 11552 gcddiv 11553 lcmgcd 11605 lcmid 11607 lcm1 11608 3lcm2e6woprm 11613 6lcm4e12 11614 mulgcddvds 11621 qredeu 11624 divgcdcoprm0 11628 divgcdcoprmex 11629 cncongr1 11630 cncongr2 11631 pw2dvdseulemle 11690 phiprmpw 11743 |
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