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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 11652 | . 2 ⊢ 1 ∈ ℕ | |
2 | nnnegz 11987 | . 2 ⊢ (1 ∈ ℕ → -1 ∈ ℤ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -1 ∈ ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 1c1 10541 -cneg 10874 ℕcn 11641 ℤcz 11984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 df-nn 11642 df-z 11985 |
This theorem is referenced by: modsumfzodifsn 13315 m1expcl 13455 risefall0lem 15383 binomfallfaclem2 15397 nthruz 15609 n2dvdsm1 15722 bitsfzo 15787 bezoutlem1 15890 pythagtriplem4 16159 odinv 18691 zrhpsgnmhm 20731 zrhpsgnelbas 20741 m2detleiblem1 21236 clmneg1 23689 plyeq0lem 24803 aaliou3lem2 24935 dvradcnv 25012 efif1olem2 25130 ang180lem3 25392 wilthimp 25652 muf 25720 ppiub 25783 lgslem2 25877 lgsfcl2 25882 lgsval2lem 25886 lgsdir2lem3 25906 lgsdir2lem4 25907 gausslemma2dlem5a 25949 gausslemma2dlem7 25952 gausslemma2d 25953 lgseisenlem2 25955 lgseisenlem4 25957 m1lgs 25967 2sqlem11 26008 2sqblem 26010 ostth3 26217 archirngz 30822 mdetpmtr1 31092 mdetpmtr12 31094 qqhval2lem 31226 bcneg1 32972 mzpsubmpt 39346 rmxm1 39537 rmym1 39538 dvradcnv2 40685 binomcxplemnotnn0 40694 cosnegpi 42154 fourierdlem24 42423 fmtnoprmfac1lem 43733 2pwp1prm 43758 lighneallem4b 43781 lighneallem4 43782 modexp2m1d 43784 41prothprmlem2 43790 |
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