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Mirrors > Home > ILE Home > Th. List > 6gcd4e2 | GIF version |
Description: The greatest common divisor of six and four is two. To calculate this gcd, a simple form of Euclid's algorithm is used: (6 gcd 4) = ((4 + 2) gcd 4) = (2 gcd 4) and (2 gcd 4) = (2 gcd (2 + 2)) = (2 gcd 2) = 2. (Contributed by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
6gcd4e2 | ⊢ (6 gcd 4) = 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 8878 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 9068 | . . 3 ⊢ 6 ∈ ℤ |
3 | 4z 9077 | . . 3 ⊢ 4 ∈ ℤ | |
4 | gcdcom 11651 | . . 3 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) = (4 gcd 6)) | |
5 | 2, 3, 4 | mp2an 422 | . 2 ⊢ (6 gcd 4) = (4 gcd 6) |
6 | 4cn 8791 | . . . 4 ⊢ 4 ∈ ℂ | |
7 | 2cn 8784 | . . . 4 ⊢ 2 ∈ ℂ | |
8 | 4p2e6 8856 | . . . 4 ⊢ (4 + 2) = 6 | |
9 | 6, 7, 8 | addcomli 7900 | . . 3 ⊢ (2 + 4) = 6 |
10 | 9 | oveq2i 5778 | . 2 ⊢ (4 gcd (2 + 4)) = (4 gcd 6) |
11 | 2z 9075 | . . . . 5 ⊢ 2 ∈ ℤ | |
12 | gcdadd 11662 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 2 ∈ ℤ) → (2 gcd 2) = (2 gcd (2 + 2))) | |
13 | 11, 11, 12 | mp2an 422 | . . . 4 ⊢ (2 gcd 2) = (2 gcd (2 + 2)) |
14 | 2p2e4 8840 | . . . . . 6 ⊢ (2 + 2) = 4 | |
15 | 14 | oveq2i 5778 | . . . . 5 ⊢ (2 gcd (2 + 2)) = (2 gcd 4) |
16 | gcdcom 11651 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → (2 gcd 4) = (4 gcd 2)) | |
17 | 11, 3, 16 | mp2an 422 | . . . . 5 ⊢ (2 gcd 4) = (4 gcd 2) |
18 | 15, 17 | eqtri 2158 | . . . 4 ⊢ (2 gcd (2 + 2)) = (4 gcd 2) |
19 | 13, 18 | eqtri 2158 | . . 3 ⊢ (2 gcd 2) = (4 gcd 2) |
20 | gcdid 11663 | . . . . 5 ⊢ (2 ∈ ℤ → (2 gcd 2) = (abs‘2)) | |
21 | 11, 20 | ax-mp 5 | . . . 4 ⊢ (2 gcd 2) = (abs‘2) |
22 | 2re 8783 | . . . . 5 ⊢ 2 ∈ ℝ | |
23 | 0le2 8803 | . . . . 5 ⊢ 0 ≤ 2 | |
24 | absid 10836 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) | |
25 | 22, 23, 24 | mp2an 422 | . . . 4 ⊢ (abs‘2) = 2 |
26 | 21, 25 | eqtri 2158 | . . 3 ⊢ (2 gcd 2) = 2 |
27 | gcdadd 11662 | . . . 4 ⊢ ((4 ∈ ℤ ∧ 2 ∈ ℤ) → (4 gcd 2) = (4 gcd (2 + 4))) | |
28 | 3, 11, 27 | mp2an 422 | . . 3 ⊢ (4 gcd 2) = (4 gcd (2 + 4)) |
29 | 19, 26, 28 | 3eqtr3ri 2167 | . 2 ⊢ (4 gcd (2 + 4)) = 2 |
30 | 5, 10, 29 | 3eqtr2i 2164 | 1 ⊢ (6 gcd 4) = 2 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 ℝcr 7612 0cc0 7613 + caddc 7616 ≤ cle 7794 2c2 8764 4c4 8766 6c6 8768 ℤcz 9047 abscabs 10762 gcd cgcd 11624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-sup 6864 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-fzo 9913 df-fl 10036 df-mod 10089 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-dvds 11483 df-gcd 11625 |
This theorem is referenced by: 6lcm4e12 11757 |
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