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Mirrors > Home > ILE Home > Th. List > nn0cni | GIF version |
Description: A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.) |
Ref | Expression |
---|---|
nn0re.1 | ⊢ 𝐴 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0cni | ⊢ 𝐴 ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
2 | 1 | nn0rei 9106 | . 2 ⊢ 𝐴 ∈ ℝ |
3 | 2 | recni 7892 | 1 ⊢ 𝐴 ∈ ℂ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 ℂcc 7732 ℕ0cn0 9095 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-sep 4084 ax-cnex 7825 ax-resscn 7826 ax-1re 7828 ax-addrcl 7831 ax-rnegex 7843 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-sn 3567 df-int 3810 df-inn 8839 df-n0 9096 |
This theorem is referenced by: nn0le2xi 9145 num0u 9310 num0h 9311 numsuc 9313 numsucc 9339 numma 9343 nummac 9344 numma2c 9345 numadd 9346 numaddc 9347 nummul1c 9348 nummul2c 9349 decrmanc 9356 decrmac 9357 decaddi 9359 decaddci 9360 decsubi 9362 decmul1 9363 decmulnc 9366 11multnc 9367 decmul10add 9368 6p5lem 9369 4t3lem 9396 7t3e21 9409 7t6e42 9412 8t3e24 9415 8t4e32 9416 8t8e64 9420 9t3e27 9422 9t4e36 9423 9t5e45 9424 9t6e54 9425 9t7e63 9426 9t11e99 9429 decbin0 9439 decbin2 9440 sq10 10597 3dec 10599 3dvdsdec 11768 3dvds2dec 11769 3lcm2e6 12050 |
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