![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gcdn0cl | Structured version Visualization version GIF version |
Description: Closure of the gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcdn0cl | ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdn0val 15554 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) | |
2 | eqid 2800 | . . . 4 ⊢ {𝑛 ∈ ℤ ∣ ∀𝑧 ∈ {𝑀, 𝑁}𝑛 ∥ 𝑧} = {𝑛 ∈ ℤ ∣ ∀𝑧 ∈ {𝑀, 𝑁}𝑛 ∥ 𝑧} | |
3 | eqid 2800 | . . . 4 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} | |
4 | 2, 3 | gcdcllem3 15557 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∈ ℕ ∧ (sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∥ 𝑀 ∧ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∥ 𝑁) ∧ ((𝐾 ∈ ℤ ∧ 𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁) → 𝐾 ≤ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )))) |
5 | 4 | simp1d 1173 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∈ ℕ) |
6 | 1, 5 | eqeltrd 2879 | 1 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∀wral 3090 {crab 3094 {cpr 4371 class class class wbr 4844 (class class class)co 6879 supcsup 8589 ℝcr 10224 0cc0 10225 < clt 10364 ≤ cle 10365 ℕcn 11313 ℤcz 11665 ∥ cdvds 15318 gcd cgcd 15550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-pre-sup 10303 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-sup 8591 df-inf 8592 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-nn 11314 df-2 11375 df-3 11376 df-n0 11580 df-z 11666 df-uz 11930 df-rp 12074 df-seq 13055 df-exp 13114 df-cj 14179 df-re 14180 df-im 14181 df-sqrt 14315 df-abs 14316 df-dvds 15319 df-gcd 15551 |
This theorem is referenced by: gcdcl 15562 gcdnncl 15563 gcd2n0cl 15565 divgcdnn 15570 gcdeq0 15572 gcd1 15583 sqgcd 15612 bezoutr1 15616 3lcm2e6woprm 15662 6lcm4e12 15663 qredeu 15705 rpdvds 15707 coprm 15755 rpexp 15764 divnumden 15788 divdenle 15789 phimullem 15816 phisum 15827 pythagtriplem4 15856 pythagtriplem8 15860 pythagtriplem9 15861 pythagtriplem19 15870 pcgcd1 15913 pc2dvds 15915 pockthlem 15941 ablfacrplem 18779 znrrg 20234 dvdsmulf1o 25271 dchrelbas4 25319 lgsne0 25411 2sqlem8a 25501 2sqlem8 25502 2sqcoprm 30162 divgcdoddALTV 42370 |
Copyright terms: Public domain | W3C validator |