Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > neg1z | Structured version Visualization version GIF version |
Description: -1 is an integer. (Contributed by David A. Wheeler, 5-Dec-2018.) |
Ref | Expression |
---|---|
neg1z | ⊢ -1 ∈ ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 11984 | . 2 ⊢ 1 ∈ ℕ | |
2 | nnnegz 12322 | . 2 ⊢ (1 ∈ ℕ → -1 ∈ ℤ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -1 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 1c1 10872 -cneg 11206 ℕcn 11973 ℤcz 12319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 df-sub 11207 df-neg 11208 df-nn 11974 df-z 12320 |
This theorem is referenced by: modsumfzodifsn 13664 m1expcl 13805 risefall0lem 15736 binomfallfaclem2 15750 nthruz 15962 n2dvdsm1 16078 bitsfzo 16142 bezoutlem1 16247 pythagtriplem4 16520 odinv 19168 zrhpsgnmhm 20789 zrhpsgnelbas 20799 m2detleiblem1 21773 clmneg1 24245 plyeq0lem 25371 aaliou3lem2 25503 dvradcnv 25580 efif1olem2 25699 ang180lem3 25961 wilthimp 26221 muf 26289 ppiub 26352 lgslem2 26446 lgsfcl2 26451 lgsval2lem 26455 lgsdir2lem3 26475 lgsdir2lem4 26476 gausslemma2dlem5a 26518 gausslemma2dlem7 26521 gausslemma2d 26522 lgseisenlem2 26524 lgseisenlem4 26526 m1lgs 26536 2sqlem11 26577 2sqblem 26579 ostth3 26786 archirngz 31443 mdetpmtr1 31773 mdetpmtr12 31775 qqhval2lem 31931 bcneg1 33702 mzpsubmpt 40565 rmxm1 40756 rmym1 40757 dvradcnv2 41965 binomcxplemnotnn0 41974 cosnegpi 43408 fourierdlem24 43672 fmtnoprmfac1lem 45016 2pwp1prm 45041 lighneallem4b 45061 lighneallem4 45062 modexp2m1d 45064 41prothprmlem2 45070 |
Copyright terms: Public domain | W3C validator |