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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 11727 | . 2 ⊢ 1 ∈ ℕ | |
2 | nnnegz 12065 | . 2 ⊢ (1 ∈ ℕ → -1 ∈ ℤ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -1 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 1c1 10616 -cneg 10949 ℕcn 11716 ℤcz 12062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-ltxr 10758 df-sub 10950 df-neg 10951 df-nn 11717 df-z 12063 |
This theorem is referenced by: modsumfzodifsn 13403 m1expcl 13544 risefall0lem 15472 binomfallfaclem2 15486 nthruz 15698 n2dvdsm1 15814 bitsfzo 15878 bezoutlem1 15983 pythagtriplem4 16256 odinv 18806 zrhpsgnmhm 20400 zrhpsgnelbas 20410 m2detleiblem1 21375 clmneg1 23834 plyeq0lem 24959 aaliou3lem2 25091 dvradcnv 25168 efif1olem2 25287 ang180lem3 25549 wilthimp 25809 muf 25877 ppiub 25940 lgslem2 26034 lgsfcl2 26039 lgsval2lem 26043 lgsdir2lem3 26063 lgsdir2lem4 26064 gausslemma2dlem5a 26106 gausslemma2dlem7 26109 gausslemma2d 26110 lgseisenlem2 26112 lgseisenlem4 26114 m1lgs 26124 2sqlem11 26165 2sqblem 26167 ostth3 26374 archirngz 31020 mdetpmtr1 31345 mdetpmtr12 31347 qqhval2lem 31501 bcneg1 33273 mzpsubmpt 40137 rmxm1 40328 rmym1 40329 dvradcnv2 41503 binomcxplemnotnn0 41512 cosnegpi 42950 fourierdlem24 43214 fmtnoprmfac1lem 44550 2pwp1prm 44575 lighneallem4b 44595 lighneallem4 44596 modexp2m1d 44598 41prothprmlem2 44604 |
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