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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 7169 | . . 3 ⊢ ((𝐾 mod 𝑁) gcd 𝑁) = ((𝑅 mod 𝑁) gcd 𝑁) |
3 | gcdi.1 | . . . . 5 ⊢ 𝐾 ∈ ℕ0 | |
4 | 3 | nn0zi 12010 | . . . 4 ⊢ 𝐾 ∈ ℤ |
5 | gcdmodi.3 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
6 | modgcd 15883 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐾 mod 𝑁) gcd 𝑁) = (𝐾 gcd 𝑁)) | |
7 | 4, 5, 6 | mp2an 690 | . . 3 ⊢ ((𝐾 mod 𝑁) gcd 𝑁) = (𝐾 gcd 𝑁) |
8 | gcdi.2 | . . . . 5 ⊢ 𝑅 ∈ ℕ0 | |
9 | 8 | nn0zi 12010 | . . . 4 ⊢ 𝑅 ∈ ℤ |
10 | modgcd 15883 | . . . 4 ⊢ ((𝑅 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑅 mod 𝑁) gcd 𝑁) = (𝑅 gcd 𝑁)) | |
11 | 9, 5, 10 | mp2an 690 | . . 3 ⊢ ((𝑅 mod 𝑁) gcd 𝑁) = (𝑅 gcd 𝑁) |
12 | 2, 7, 11 | 3eqtr3i 2855 | . 2 ⊢ (𝐾 gcd 𝑁) = (𝑅 gcd 𝑁) |
13 | 5 | nnzi 12009 | . . 3 ⊢ 𝑁 ∈ ℤ |
14 | gcdcom 15865 | . . 3 ⊢ ((𝑅 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑅 gcd 𝑁) = (𝑁 gcd 𝑅)) | |
15 | 9, 13, 14 | mp2an 690 | . 2 ⊢ (𝑅 gcd 𝑁) = (𝑁 gcd 𝑅) |
16 | gcdmodi.5 | . 2 ⊢ (𝑁 gcd 𝑅) = 𝐺 | |
17 | 12, 15, 16 | 3eqtri 2851 | 1 ⊢ (𝐾 gcd 𝑁) = 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 (class class class)co 7159 ℕcn 11641 ℕ0cn0 11900 ℤcz 11984 mod cmo 13240 gcd cgcd 15846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-dvds 15611 df-gcd 15847 |
This theorem is referenced by: 1259lem5 16471 2503lem3 16475 4001lem4 16480 |
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