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Mirrors > Home > MPE Home > Th. List > uznn0sub | Structured version Visualization version GIF version |
Description: The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.) |
Ref | Expression |
---|---|
uznn0sub | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 11974 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
2 | znn0sub 11752 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0)) | |
3 | 2 | biimp3a 1599 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0) |
4 | 1, 3 | sylbi 209 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 ∈ wcel 2166 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ≤ cle 10392 − cmin 10585 ℕ0cn0 11618 ℤcz 11704 ℤ≥cuz 11968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 |
This theorem is referenced by: fznn0sub 12666 elfzmlbm 12744 fzen2 13063 fzfi 13066 leexp2a 13210 hashfz 13503 ccatrn 13649 swrds2m 14062 isercoll2 14776 iseralt 14792 cvgrat 14988 fprodser 15052 dvdsexp 15426 bitsshft 15570 prmm2nn0 15782 hashdvds 15851 prmdiv 15861 prmdiveq 15862 pcaddlem 15963 vdwlem5 16060 vdwlem8 16063 dvn2bss 24092 aaliou3lem2 24497 lgsquadlem1 25518 crctcshwlkn0lem5 27113 clwwlknonex2lem1 27482 clwwlknonex2lem2 27483 numclwlk1lem2 27773 numclwwlk3lem1 27797 fiblem 31006 signstfveq0 31202 signstfveq0OLD 31203 irrapxlem3 38232 itgsinexp 40965 wallispilem3 41078 fmtnorec3 42290 fmtnorec4 42291 |
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