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| Mirrors > Home > MPE Home > Th. List > gcdcomd | Structured version Visualization version GIF version | ||
| Description: The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.) |
| Ref | Expression |
|---|---|
| gcdcomd.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| gcdcomd.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| Ref | Expression |
|---|---|
| gcdcomd | ⊢ (𝜑 → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | gcdcomd.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 2 | gcdcomd.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
| 3 | gcdcom 16550 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℤcz 12613 gcd cgcd 16531 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-gcd 16532 |
| This theorem is referenced by: modgcd 16569 rplpwr 16595 rprpwr 16596 coprmprod 16698 rpexp12i 16761 phiprmpw 16813 eulerthlem1 16818 eulerthlem2 16819 prmdiv 16822 coprimeprodsq 16846 pythagtriplem3 16856 prmpwdvds 16942 prmgaplem7 17095 gexexlem 19870 ablfacrp2 20087 pgpfac1lem2 20095 mpodvdsmulf1o 27237 dvdsmulf1o 27239 perfect1 27272 perfectlem1 27273 lgslem1 27341 lgsqrlem2 27391 lgsqr 27395 gausslemma2dlem0c 27402 lgsquad2lem2 27429 lgsquad2 27430 lgsquad3 27431 2sqlem8 27470 2sqmod 27480 nn0prpwlem 36323 aks4d1p8d2 42086 aks4d1p8d3 42087 hashscontpow1 42122 aks6d1c4 42125 aks5 42205 fltbccoprm 42651 flt4lem3 42658 flt4lem5c 42664 flt4lem5d 42665 flt4lem5e 42666 flt4lem5f 42667 flt4lem7 42669 nna4b4nsq 42670 jm2.19lem2 43002 jm2.20nn 43009 perfectALTVlem1 47708 |
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