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Mirrors > Home > ILE Home > Th. List > gcdcom | 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 262 | . . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑁 = 0 ∧ 𝑀 = 0)) | |
2 | ancom 262 | . . . . . 6 ⊢ ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)) | |
3 | 2 | a1i 9 | . . . . 5 ⊢ (𝑛 ∈ ℤ → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀))) |
4 | 3 | rabbiia 2596 | . . . 4 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} = {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)} |
5 | 4 | supeq1i 6495 | . . 3 ⊢ sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑁 ∧ 𝑛 ∥ 𝑀)}, ℝ, < ) |
6 | 1, 5 | ifbieq2i 3389 | . 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 10558 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))) | |
8 | gcdval 10558 | . . 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 264 | . 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 2141 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 {crab 2357 ifcif 3368 class class class wbr 3805 (class class class)co 5563 supcsup 6489 ℝcr 7094 0cc0 7095 < clt 7267 ℤcz 8484 ∥ cdvds 10403 gcd cgcd 10545 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 ax-cnex 7181 ax-resscn 7182 ax-1cn 7183 ax-1re 7184 ax-icn 7185 ax-addcl 7186 ax-addrcl 7187 ax-mulcl 7188 ax-mulrcl 7189 ax-addcom 7190 ax-mulcom 7191 ax-addass 7192 ax-mulass 7193 ax-distr 7194 ax-i2m1 7195 ax-0lt1 7196 ax-1rid 7197 ax-0id 7198 ax-rnegex 7199 ax-precex 7200 ax-cnre 7201 ax-pre-ltirr 7202 ax-pre-ltwlin 7203 ax-pre-lttrn 7204 ax-pre-apti 7205 ax-pre-ltadd 7206 ax-pre-mulgt0 7207 ax-pre-mulext 7208 ax-arch 7209 ax-caucvg 7210 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-if 3369 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-id 4076 df-po 4079 df-iso 4080 df-iord 4149 df-on 4151 df-ilim 4152 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-riota 5519 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-1st 5818 df-2nd 5819 df-recs 5974 df-frec 6060 df-sup 6491 df-pnf 7269 df-mnf 7270 df-xr 7271 df-ltxr 7272 df-le 7273 df-sub 7400 df-neg 7401 df-reap 7794 df-ap 7801 df-div 7880 df-inn 8159 df-2 8217 df-3 8218 df-4 8219 df-n0 8408 df-z 8485 df-uz 8753 df-q 8838 df-rp 8868 df-fz 9158 df-fzo 9282 df-fl 9404 df-mod 9457 df-iseq 9574 df-iexp 9625 df-cj 9930 df-re 9931 df-im 9932 df-rsqrt 10085 df-abs 10086 df-dvds 10404 df-gcd 10546 |
This theorem is referenced by: divgcdnnr 10574 gcdid0 10578 neggcd 10581 gcdabs2 10588 modgcd 10589 1gcd 10590 6gcd4e2 10591 rplpwr 10623 rppwr 10624 eucalginv 10645 3lcm2e6woprm 10675 coprmdvds 10681 qredeq 10685 divgcdcoprmex 10691 cncongr1 10692 rpexp12i 10741 cncongrprm 10743 phiprmpw 10805 ex-gcd 10828 |
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