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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 16546 | . 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 1536 ∈ wcel 2105 (class class class)co 7430 ℤcz 12610 gcd cgcd 16527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-mulcl 11214 ax-i2m1 11220 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-gcd 16528 |
This theorem is referenced by: modgcd 16565 rplpwr 16591 rprpwr 16592 coprmprod 16694 rpexp12i 16757 phiprmpw 16809 eulerthlem1 16814 eulerthlem2 16815 prmdiv 16818 coprimeprodsq 16841 pythagtriplem3 16851 prmpwdvds 16937 prmgaplem7 17090 gexexlem 19884 ablfacrp2 20101 pgpfac1lem2 20109 mpodvdsmulf1o 27251 dvdsmulf1o 27253 perfect1 27286 perfectlem1 27287 lgslem1 27355 lgsqrlem2 27405 lgsqr 27409 gausslemma2dlem0c 27416 lgsquad2lem2 27443 lgsquad2 27444 lgsquad3 27445 2sqlem8 27484 2sqmod 27494 nn0prpwlem 36304 aks4d1p8d2 42066 aks4d1p8d3 42067 hashscontpow1 42102 aks6d1c4 42105 aks5 42185 fltbccoprm 42627 flt4lem3 42634 flt4lem5c 42640 flt4lem5d 42641 flt4lem5e 42642 flt4lem5f 42643 flt4lem7 42645 nna4b4nsq 42646 jm2.19lem2 42978 jm2.20nn 42985 perfectALTVlem1 47645 |
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