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Mirrors > Home > MPE Home > Th. List > gcdcom | Structured version Visualization version GIF version |
Description: The gcd operator is commutative. Theorem 1.4(a) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
gcdcom | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 463 | . . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑁 = 0 ∧ 𝑀 = 0)) | |
2 | ancom 463 | . . . . 5 ⊢ ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)) | |
3 | 2 | rabbii 3473 | . . . 4 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)} |
4 | 3 | supeq1i 8911 | . . 3 ⊢ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)}, ℝ, < ) |
5 | 1, 4 | ifbieq2i 4491 | . 2 ⊢ if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < )) = if((𝑁 = 0 ∧ 𝑀 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)}, ℝ, < )) |
6 | gcdval 15845 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))) | |
7 | gcdval 15845 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = if((𝑁 = 0 ∧ 𝑀 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)}, ℝ, < ))) | |
8 | 7 | ancoms 461 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 gcd 𝑀) = if((𝑁 = 0 ∧ 𝑀 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)}, ℝ, < ))) |
9 | 5, 6, 8 | 3eqtr4a 2882 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 ifcif 4467 class class class wbr 5066 (class class class)co 7156 supcsup 8904 ℝcr 10536 0cc0 10537 < clt 10675 ℤcz 11982 ∥ cdvds 15607 gcd cgcd 15843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-i2m1 10605 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-gcd 15844 |
This theorem is referenced by: divgcdnnr 15864 gcdid0 15868 neggcd 15871 gcdabs2 15879 modgcd 15880 1gcd 15881 6gcd4e2 15886 rplpwr 15907 rppwr 15908 eucalginv 15928 3lcm2e6woprm 15959 coprmdvds 15997 qredeq 16001 coprmprod 16005 divgcdcoprmex 16010 cncongr1 16011 rpexp12i 16066 cncongrprm 16069 phiprmpw 16113 eulerthlem1 16118 eulerthlem2 16119 fermltl 16121 prmdiv 16122 vfermltl 16138 coprimeprodsq 16145 coprimeprodsq2 16146 pythagtriplem3 16155 pythagtrip 16171 pcgcd 16214 prmpwdvds 16240 pockthlem 16241 prmgaplem7 16393 gcdi 16409 gcdmodi 16410 1259lem5 16468 2503lem3 16472 4001lem4 16477 odinv 18688 gexexlem 18972 ablfacrp2 19189 pgpfac1lem2 19197 dvdsmulf1o 25771 perfect1 25804 perfectlem1 25805 lgslem1 25873 lgsprme0 25915 lgsdirnn0 25920 lgsqrlem2 25923 lgsqr 25927 gausslemma2dlem0c 25934 lgsquad2lem2 25961 lgsquad2 25962 lgsquad3 25963 2sqlem8 26002 2sqmod 26012 ex-gcd 28236 gcd32 32983 nn0prpwlem 33670 jm2.19lem2 39607 jm2.20nn 39614 goldbachthlem2 43728 goldbachth 43729 gcd2odd1 43853 perfectALTVlem1 43906 fpprwpprb 43925 |
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