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Mirrors > Home > MPE Home > Th. List > nnabscl | Structured version Visualization version GIF 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 | ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zabscl 14842 | . . 3 ⊢ (𝑁 ∈ ℤ → (abs‘𝑁) ∈ ℤ) | |
2 | 1 | adantr 484 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℤ) |
3 | zcn 12146 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
4 | absgt0 14853 | . . . 4 ⊢ (𝑁 ∈ ℂ → (𝑁 ≠ 0 ↔ 0 < (abs‘𝑁))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 ≠ 0 ↔ 0 < (abs‘𝑁))) |
6 | 5 | biimpa 480 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 0 < (abs‘𝑁)) |
7 | elnnz 12151 | . 2 ⊢ ((abs‘𝑁) ∈ ℕ ↔ ((abs‘𝑁) ∈ ℤ ∧ 0 < (abs‘𝑁))) | |
8 | 2, 6, 7 | sylanbrc 586 | 1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2112 ≠ wne 2932 class class class wbr 5039 ‘cfv 6358 ℂcc 10692 0cc0 10694 < clt 10832 ℕcn 11795 ℤcz 12141 abscabs 14762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 |
This theorem is referenced by: dvdsleabs 15835 divalglem1 15918 divalglem5 15921 divalglem7 15923 divalglem8 15924 divalglem9 15925 bezoutlem1 16062 gcdmultiplezOLD 16076 dvdssq 16087 lcmcllem 16116 lcmgcd 16127 lcmdvds 16128 absprodnn 16138 pc2dvds 16395 zringlpirlem1 20403 aalioulem4 25182 lgscllem 26139 lgsneg 26156 lgsdir 26167 lgsdilem2 26168 lgsdi 26169 lgsne0 26170 lgsqr 26186 dffltz 40115 pellexlem6 40300 pellex 40301 dvdsabsmod0 40453 jm2.19 40459 etransc 43442 |
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