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Mirrors > Home > MPE Home > Th. List > nnabscl | Structured version Unicode version |
Description: The absolute value of a nonzero integer is a positive integer. (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
nnabscl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0abscl 12922 |
. . . 4
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2 | 1 | nn0zd 10859 |
. . 3
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3 | 2 | adantr 465 |
. 2
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4 | zcn 10765 |
. . . 4
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5 | absgt0 12933 |
. . . 4
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6 | 4, 5 | syl 16 |
. . 3
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7 | 6 | biimpa 484 |
. 2
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8 | elnnz 10770 |
. 2
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9 | 3, 7, 8 | sylanbrc 664 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-cnex 9452 ax-resscn 9453 ax-1cn 9454 ax-icn 9455 ax-addcl 9456 ax-addrcl 9457 ax-mulcl 9458 ax-mulrcl 9459 ax-mulcom 9460 ax-addass 9461 ax-mulass 9462 ax-distr 9463 ax-i2m1 9464 ax-1ne0 9465 ax-1rid 9466 ax-rnegex 9467 ax-rrecex 9468 ax-cnre 9469 ax-pre-lttri 9470 ax-pre-lttrn 9471 ax-pre-ltadd 9472 ax-pre-mulgt0 9473 ax-pre-sup 9474 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-2nd 6691 df-recs 6945 df-rdg 6979 df-er 7214 df-en 7424 df-dom 7425 df-sdom 7426 df-sup 7805 df-pnf 9534 df-mnf 9535 df-xr 9536 df-ltxr 9537 df-le 9538 df-sub 9711 df-neg 9712 df-div 10108 df-nn 10437 df-2 10494 df-3 10495 df-n0 10694 df-z 10761 df-uz 10976 df-rp 11106 df-seq 11927 df-exp 11986 df-cj 12709 df-re 12710 df-im 12711 df-sqr 12845 df-abs 12846 |
This theorem is referenced by: dvdsleabs 13700 divalglem1 13719 divalglem5 13722 divalglem7 13724 divalglem8 13725 divalglem9 13726 bezoutlem1 13843 gcdmultiplez 13856 dvdssq 13865 pc2dvds 14066 zringlpirlem1 18031 zlpirlem1 18036 aalioulem4 21937 lgscllem 22778 lgsneg 22794 lgsdir 22805 lgsdilem2 22806 lgsdi 22807 lgsne0 22808 lgsqr 22821 pellexlem6 29343 pellex 29344 dvdsabsmod0 29503 jm2.19 29510 |
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