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Mirrors > Home > ILE Home > Th. List > nnzd | GIF version |
Description: A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
nnzd.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnzd | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnzd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | 1 | nnnn0d 8636 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
3 | 2 | nn0zd 8776 | 1 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1436 ℕcn 8334 ℤcz 8660 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-addass 7368 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-0id 7374 ax-rnegex 7375 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-ltadd 7382 |
This theorem depends on definitions: df-bi 115 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-br 3815 df-opab 3869 df-id 4087 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-iota 4937 df-fun 4974 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-inn 8335 df-n0 8584 df-z 8661 |
This theorem is referenced by: qapne 9033 qtri3or 9557 exbtwnzlemstep 9562 modifeq2int 9696 modsumfzodifsn 9706 addmodlteq 9708 expnegap0 9814 expaddzaplem 9849 expmulzap 9852 facndiv 9996 bcval 10006 ibcval5 10020 bcpasc 10023 caucvgre 10255 cvg1nlemcau 10258 cvg1nlemres 10259 resqrexlemdecn 10286 resqrexlemnmsq 10291 resqrexlemnm 10292 resqrexlemcvg 10293 resqrexlemoverl 10295 sumeq2d 10584 sumeq2 10585 dvdsle 10639 fzm1ndvds 10651 dvdsfac 10655 dvdsmod 10657 divalglemeunn 10715 gcddvds 10749 gcdnncl 10753 gcd1 10772 bezoutlemnewy 10779 bezoutlemstep 10780 mulgcd 10799 gcdmultiplez 10804 rplpwr 10810 rppwr 10811 sqgcd 10812 dvdssq 10814 lcmneg 10850 lcmgcdlem 10853 ncoprmgcdne1b 10865 rpdvds 10875 congr 10876 cncongr1 10879 cncongr2 10880 prmz 10887 prmind2 10896 divgcdodd 10916 isprm6 10920 prmexpb 10924 prmfac1 10925 rpexp 10926 sqrt2irrlem 10934 pw2dvdslemn 10937 pw2dvdseulemle 10939 oddpwdclemxy 10941 oddpwdclemodd 10944 sqpweven 10947 2sqpwodd 10948 sqrt2irraplemnn 10951 numdensq 10974 phivalfi 10982 hashdvds 10991 phiprmpw 10992 crth 10994 phimullem 10995 hashgcdlem 10997 hashgcdeq 10998 oddennn 10999 |
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