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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 465 | . . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑁 = 0 ∧ 𝑀 = 0)) | |
2 | ancom 465 | . . . . . 6 ⊢ ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)) | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝑛 ∈ ℤ → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀))) |
4 | 3 | rabbiia 3161 | . . . 4 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)} |
5 | 4 | supeq1i 8236 | . . 3 ⊢ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)}, ℝ, < ) |
6 | 1, 5 | ifbieq2i 4060 | . 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, < )) |
7 | gcdval 15056 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))) | |
8 | gcdval 15056 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = if((𝑁 = 0 ∧ 𝑀 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)}, ℝ, < ))) | |
9 | 8 | ancoms 468 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 gcd 𝑀) = if((𝑁 = 0 ∧ 𝑀 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)}, ℝ, < ))) |
10 | 6, 7, 9 | 3eqtr4a 2670 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ifcif 4036 class class class wbr 4583 (class class class)co 6549 supcsup 8229 ℝcr 9814 0cc0 9815 < clt 9953 ℤcz 11254 ∥ cdvds 14821 gcd cgcd 15054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-gcd 15055 |
This theorem is referenced by: divgcdnnr 15075 gcdid0 15079 neggcd 15082 gcdabs2 15090 modgcd 15091 1gcd 15092 6gcd4e2 15093 rplpwr 15114 rppwr 15115 eucalginv 15135 3lcm2e6woprm 15166 coprmdvds 15204 qredeq 15209 coprmprod 15213 divgcdcoprmex 15218 cncongr1 15219 rpexp12i 15272 cncongrprm 15275 phiprmpw 15319 eulerthlem1 15324 eulerthlem2 15325 fermltl 15327 prmdiv 15328 vfermltl 15344 coprimeprodsq 15351 coprimeprodsq2 15352 pythagtriplem3 15361 pythagtrip 15377 pcgcd 15420 prmpwdvds 15446 pockthlem 15447 prmgaplem7 15599 gcdi 15615 gcdmodi 15616 1259lem5 15680 2503lem3 15684 4001lem4 15689 odinv 17801 gexexlem 18078 ablfacrp2 18289 pgpfac1lem2 18297 dvdsmulf1o 24720 perfect1 24753 perfectlem1 24754 lgslem1 24822 lgsprme0 24864 lgsdirnn0 24869 lgsqrlem2 24872 lgsqr 24876 gausslemma2dlem0c 24883 lgsquad2lem2 24910 lgsquad2 24911 lgsquad3 24912 2sqlem8 24951 ex-gcd 26706 2sqmod 28979 gcd32 30890 nn0prpwlem 31487 jm2.19lem2 36575 jm2.20nn 36582 goldbachthlem2 39996 goldbachth 39997 perfectALTVlem1 40164 |
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