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Mirrors > Home > MPE Home > Th. List > gcddvds | Structured version Visualization version GIF version |
Description: The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcddvds | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11716 | . . . . . 6 ⊢ 0 ∈ ℤ | |
2 | dvds0 15375 | . . . . . 6 ⊢ (0 ∈ ℤ → 0 ∥ 0) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 0 ∥ 0 |
4 | breq2 4878 | . . . . . . 7 ⊢ (𝑀 = 0 → (0 ∥ 𝑀 ↔ 0 ∥ 0)) | |
5 | breq2 4878 | . . . . . . 7 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0)) | |
6 | 4, 5 | bi2anan9 631 | . . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) ↔ (0 ∥ 0 ∧ 0 ∥ 0))) |
7 | anidm 562 | . . . . . 6 ⊢ ((0 ∥ 0 ∧ 0 ∥ 0) ↔ 0 ∥ 0) | |
8 | 6, 7 | syl6bb 279 | . . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) ↔ 0 ∥ 0)) |
9 | 3, 8 | mpbiri 250 | . . . 4 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)) |
10 | oveq12 6915 | . . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0)) | |
11 | gcd0val 15593 | . . . . . . 7 ⊢ (0 gcd 0) = 0 | |
12 | 10, 11 | syl6eq 2878 | . . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = 0) |
13 | 12 | breq1d 4884 | . . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ 0 ∥ 𝑀)) |
14 | 12 | breq1d 4884 | . . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ 0 ∥ 𝑁)) |
15 | 13, 14 | anbi12d 626 | . . . 4 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ↔ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁))) |
16 | 9, 15 | mpbird 249 | . . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) |
17 | 16 | adantl 475 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) |
18 | eqid 2826 | . . . . 5 ⊢ {𝑛 ∈ ℤ ∣ ∀𝑧 ∈ {𝑀, 𝑁}𝑛 ∥ 𝑧} = {𝑛 ∈ ℤ ∣ ∀𝑧 ∈ {𝑀, 𝑁}𝑛 ∥ 𝑧} | |
19 | eqid 2826 | . . . . 5 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} | |
20 | 18, 19 | gcdcllem3 15597 | . . . 4 ⊢ (((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, < )))) |
21 | 20 | simp2d 1179 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∥ 𝑀 ∧ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∥ 𝑁)) |
22 | gcdn0val 15594 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) | |
23 | 22 | breq1d 4884 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∥ 𝑀)) |
24 | 22 | breq1d 4884 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∥ 𝑁)) |
25 | 23, 24 | anbi12d 626 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ↔ (sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∥ 𝑀 ∧ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∥ 𝑁))) |
26 | 21, 25 | mpbird 249 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) |
27 | 17, 26 | pm2.61dan 849 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3118 {crab 3122 {cpr 4400 class class class wbr 4874 (class class class)co 6906 supcsup 8616 ℝcr 10252 0cc0 10253 < clt 10392 ≤ cle 10393 ℕcn 11351 ℤcz 11705 ∥ cdvds 15358 gcd cgcd 15590 |
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 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
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 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-seq 13097 df-exp 13156 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-dvds 15359 df-gcd 15591 |
This theorem is referenced by: zeqzmulgcd 15606 divgcdz 15607 divgcdnn 15610 gcd0id 15614 gcdneg 15617 gcdaddmlem 15619 gcd1 15623 bezoutlem4 15633 dvdsgcdb 15636 dfgcd2 15637 mulgcd 15639 gcdzeq 15645 dvdsmulgcd 15648 sqgcd 15652 dvdssqlem 15653 bezoutr 15655 gcddvdslcm 15689 lcmgcdlem 15693 lcmgcdeq 15699 coprmgcdb 15736 mulgcddvds 15742 rpmulgcd2 15743 qredeu 15745 rpdvds 15747 divgcdcoprm0 15752 divgcdodd 15794 coprm 15795 rpexp 15804 divnumden 15828 phimullem 15856 hashgcdlem 15865 hashgcdeq 15866 phisum 15867 pythagtriplem4 15896 pythagtriplem19 15910 pcgcd1 15953 pc2dvds 15955 pockthlem 15981 odmulg 18325 odadd1 18605 odadd2 18606 znunit 20272 znrrg 20274 dvdsmulf1o 25334 2sqlem8 25565 2sqcoprm 30193 qqhval2lem 30571 goldbachthlem2 42289 divgcdoddALTV 42424 |
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