![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gcdmodi | Structured version Visualization version GIF version |
Description: Calculate a GCD via Euclid's algorithm. Theorem 5.6 in [ApostolNT] p. 109. (Contributed by Mario Carneiro, 19-Feb-2014.) |
Ref | Expression |
---|---|
gcdi.1 | ⊢ 𝐾 ∈ ℕ0 |
gcdi.2 | ⊢ 𝑅 ∈ ℕ0 |
gcdmodi.3 | ⊢ 𝑁 ∈ ℕ |
gcdmodi.4 | ⊢ (𝐾 mod 𝑁) = (𝑅 mod 𝑁) |
gcdmodi.5 | ⊢ (𝑁 gcd 𝑅) = 𝐺 |
Ref | Expression |
---|---|
gcdmodi | ⊢ (𝐾 gcd 𝑁) = 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdmodi.4 | . . . 4 ⊢ (𝐾 mod 𝑁) = (𝑅 mod 𝑁) | |
2 | 1 | oveq1i 7372 | . . 3 ⊢ ((𝐾 mod 𝑁) gcd 𝑁) = ((𝑅 mod 𝑁) gcd 𝑁) |
3 | gcdi.1 | . . . . 5 ⊢ 𝐾 ∈ ℕ0 | |
4 | 3 | nn0zi 12535 | . . . 4 ⊢ 𝐾 ∈ ℤ |
5 | gcdmodi.3 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
6 | modgcd 16420 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐾 mod 𝑁) gcd 𝑁) = (𝐾 gcd 𝑁)) | |
7 | 4, 5, 6 | mp2an 691 | . . 3 ⊢ ((𝐾 mod 𝑁) gcd 𝑁) = (𝐾 gcd 𝑁) |
8 | gcdi.2 | . . . . 5 ⊢ 𝑅 ∈ ℕ0 | |
9 | 8 | nn0zi 12535 | . . . 4 ⊢ 𝑅 ∈ ℤ |
10 | modgcd 16420 | . . . 4 ⊢ ((𝑅 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑅 mod 𝑁) gcd 𝑁) = (𝑅 gcd 𝑁)) | |
11 | 9, 5, 10 | mp2an 691 | . . 3 ⊢ ((𝑅 mod 𝑁) gcd 𝑁) = (𝑅 gcd 𝑁) |
12 | 2, 7, 11 | 3eqtr3i 2773 | . 2 ⊢ (𝐾 gcd 𝑁) = (𝑅 gcd 𝑁) |
13 | 5 | nnzi 12534 | . . 3 ⊢ 𝑁 ∈ ℤ |
14 | gcdcom 16400 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑅 gcd 𝑁) = (𝑁 gcd 𝑅)) | |
15 | 9, 13, 14 | mp2an 691 | . 2 ⊢ (𝑅 gcd 𝑁) = (𝑁 gcd 𝑅) |
16 | gcdmodi.5 | . 2 ⊢ (𝑁 gcd 𝑅) = 𝐺 | |
17 | 12, 15, 16 | 3eqtri 2769 | 1 ⊢ (𝐾 gcd 𝑁) = 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7362 ℕcn 12160 ℕ0cn0 12420 ℤcz 12506 mod cmo 13781 gcd cgcd 16381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-gcd 16382 |
This theorem is referenced by: 1259lem5 17014 2503lem3 17018 4001lem4 17023 |
Copyright terms: Public domain | W3C validator |