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Mirrors > Home > MPE Home > Th. List > gcdi | Structured version Visualization version GIF version |
Description: Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.) |
Ref | Expression |
---|---|
gcdi.1 | ⊢ 𝐾 ∈ ℕ0 |
gcdi.2 | ⊢ 𝑅 ∈ ℕ0 |
gcdi.3 | ⊢ 𝑁 ∈ ℕ0 |
gcdi.5 | ⊢ (𝑁 gcd 𝑅) = 𝐺 |
gcdi.4 | ⊢ ((𝐾 · 𝑁) + 𝑅) = 𝑀 |
Ref | Expression |
---|---|
gcdi | ⊢ (𝑀 gcd 𝑁) = 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdi.1 | . . . . . . 7 ⊢ 𝐾 ∈ ℕ0 | |
2 | gcdi.3 | . . . . . . 7 ⊢ 𝑁 ∈ ℕ0 | |
3 | 1, 2 | nn0mulcli 12128 | . . . . . 6 ⊢ (𝐾 · 𝑁) ∈ ℕ0 |
4 | 3 | nn0cni 12102 | . . . . 5 ⊢ (𝐾 · 𝑁) ∈ ℂ |
5 | gcdi.2 | . . . . . 6 ⊢ 𝑅 ∈ ℕ0 | |
6 | 5 | nn0cni 12102 | . . . . 5 ⊢ 𝑅 ∈ ℂ |
7 | gcdi.4 | . . . . 5 ⊢ ((𝐾 · 𝑁) + 𝑅) = 𝑀 | |
8 | 4, 6, 7 | addcomli 11024 | . . . 4 ⊢ (𝑅 + (𝐾 · 𝑁)) = 𝑀 |
9 | 8 | oveq2i 7224 | . . 3 ⊢ (𝑁 gcd (𝑅 + (𝐾 · 𝑁))) = (𝑁 gcd 𝑀) |
10 | 1 | nn0zi 12202 | . . . 4 ⊢ 𝐾 ∈ ℤ |
11 | 2 | nn0zi 12202 | . . . 4 ⊢ 𝑁 ∈ ℤ |
12 | 5 | nn0zi 12202 | . . . 4 ⊢ 𝑅 ∈ ℤ |
13 | gcdaddm 16084 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 gcd 𝑅) = (𝑁 gcd (𝑅 + (𝐾 · 𝑁)))) | |
14 | 10, 11, 12, 13 | mp3an 1463 | . . 3 ⊢ (𝑁 gcd 𝑅) = (𝑁 gcd (𝑅 + (𝐾 · 𝑁))) |
15 | 1, 2, 5 | numcl 12306 | . . . . . 6 ⊢ ((𝐾 · 𝑁) + 𝑅) ∈ ℕ0 |
16 | 7, 15 | eqeltrri 2835 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
17 | 16 | nn0zi 12202 | . . . 4 ⊢ 𝑀 ∈ ℤ |
18 | gcdcom 16072 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) | |
19 | 17, 11, 18 | mp2an 692 | . . 3 ⊢ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀) |
20 | 9, 14, 19 | 3eqtr4i 2775 | . 2 ⊢ (𝑁 gcd 𝑅) = (𝑀 gcd 𝑁) |
21 | gcdi.5 | . 2 ⊢ (𝑁 gcd 𝑅) = 𝐺 | |
22 | 20, 21 | eqtr3i 2767 | 1 ⊢ (𝑀 gcd 𝑁) = 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 (class class class)co 7213 + caddc 10732 · cmul 10734 ℕ0cn0 12090 ℤcz 12176 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: 1259lem5 16688 2503lem3 16692 4001lem4 16697 |
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