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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 11789 | . . . . . 6 ⊢ (𝐾 · 𝑁) ∈ ℕ0 |
4 | 3 | nn0cni 11763 | . . . . 5 ⊢ (𝐾 · 𝑁) ∈ ℂ |
5 | gcdi.2 | . . . . . 6 ⊢ 𝑅 ∈ ℕ0 | |
6 | 5 | nn0cni 11763 | . . . . 5 ⊢ 𝑅 ∈ ℂ |
7 | gcdi.4 | . . . . 5 ⊢ ((𝐾 · 𝑁) + 𝑅) = 𝑀 | |
8 | 4, 6, 7 | addcomli 10685 | . . . 4 ⊢ (𝑅 + (𝐾 · 𝑁)) = 𝑀 |
9 | 8 | oveq2i 7034 | . . 3 ⊢ (𝑁 gcd (𝑅 + (𝐾 · 𝑁))) = (𝑁 gcd 𝑀) |
10 | 1 | nn0zi 11861 | . . . 4 ⊢ 𝐾 ∈ ℤ |
11 | 2 | nn0zi 11861 | . . . 4 ⊢ 𝑁 ∈ ℤ |
12 | 5 | nn0zi 11861 | . . . 4 ⊢ 𝑅 ∈ ℤ |
13 | gcdaddm 15710 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 gcd 𝑅) = (𝑁 gcd (𝑅 + (𝐾 · 𝑁)))) | |
14 | 10, 11, 12, 13 | mp3an 1453 | . . 3 ⊢ (𝑁 gcd 𝑅) = (𝑁 gcd (𝑅 + (𝐾 · 𝑁))) |
15 | 1, 2, 5 | numcl 11965 | . . . . . 6 ⊢ ((𝐾 · 𝑁) + 𝑅) ∈ ℕ0 |
16 | 7, 15 | eqeltrri 2882 | . . . . 5 ⊢ 𝑀 ∈ ℕ0 |
17 | 16 | nn0zi 11861 | . . . 4 ⊢ 𝑀 ∈ ℤ |
18 | gcdcom 15699 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)) | |
19 | 17, 11, 18 | mp2an 688 | . . 3 ⊢ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀) |
20 | 9, 14, 19 | 3eqtr4i 2831 | . 2 ⊢ (𝑁 gcd 𝑅) = (𝑀 gcd 𝑁) |
21 | gcdi.5 | . 2 ⊢ (𝑁 gcd 𝑅) = 𝐺 | |
22 | 20, 21 | eqtr3i 2823 | 1 ⊢ (𝑀 gcd 𝑁) = 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1525 ∈ wcel 2083 (class class class)co 7023 + caddc 10393 · cmul 10395 ℕ0cn0 11751 ℤcz 11835 gcd cgcd 15680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-sup 8759 df-inf 8760 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-z 11836 df-uz 12098 df-rp 12244 df-seq 13224 df-exp 13284 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-dvds 15445 df-gcd 15681 |
This theorem is referenced by: 1259lem5 16301 2503lem3 16305 4001lem4 16310 |
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