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Mirrors > Home > MPE Home > Th. List > gcdid | Structured version Visualization version GIF version |
Description: The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.) |
Ref | Expression |
---|---|
gcdid | ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 𝑁) = (abs‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 12207 | . . 3 ⊢ 1 ∈ ℤ | |
2 | 0z 12187 | . . 3 ⊢ 0 ∈ ℤ | |
3 | gcdaddm 16084 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 gcd 0) = (𝑁 gcd (0 + (1 · 𝑁)))) | |
4 | 1, 2, 3 | mp3an13 1454 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 0) = (𝑁 gcd (0 + (1 · 𝑁)))) |
5 | gcdid0 16079 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 0) = (abs‘𝑁)) | |
6 | zcn 12181 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
7 | mulid2 10832 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (1 · 𝑁) = 𝑁) | |
8 | 7 | oveq2d 7229 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (0 + (1 · 𝑁)) = (0 + 𝑁)) |
9 | addid2 11015 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (0 + 𝑁) = 𝑁) | |
10 | 8, 9 | eqtrd 2777 | . . . 4 ⊢ (𝑁 ∈ ℂ → (0 + (1 · 𝑁)) = 𝑁) |
11 | 6, 10 | syl 17 | . . 3 ⊢ (𝑁 ∈ ℤ → (0 + (1 · 𝑁)) = 𝑁) |
12 | 11 | oveq2d 7229 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd (0 + (1 · 𝑁))) = (𝑁 gcd 𝑁)) |
13 | 4, 5, 12 | 3eqtr3rd 2786 | 1 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 𝑁) = (abs‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 ℤcz 12176 abscabs 14797 gcd cgcd 16053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-dvds 15816 df-gcd 16054 |
This theorem is referenced by: 6gcd4e2 16098 gcdmultipleOLD 16112 lcmid 16166 lcmgcdeq 16169 3lcm2e6woprm 16172 phibndlem 16323 coprimeprodsq 16361 logbgcd1irr 25677 ex-gcd 28540 gcdabsorb 33434 gcdnn0id 40037 |
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