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Mirrors > Home > MPE Home > Th. List > 6gcd4e2 | Structured version Visualization version 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 11714 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 11994 | . . 3 ⊢ 6 ∈ ℤ |
3 | 4z 12004 | . . 3 ⊢ 4 ∈ ℤ | |
4 | gcdcom 15850 | . . 3 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) = (4 gcd 6)) | |
5 | 2, 3, 4 | mp2an 688 | . 2 ⊢ (6 gcd 4) = (4 gcd 6) |
6 | 4cn 11710 | . . . 4 ⊢ 4 ∈ ℂ | |
7 | 2cn 11700 | . . . 4 ⊢ 2 ∈ ℂ | |
8 | 4p2e6 11778 | . . . 4 ⊢ (4 + 2) = 6 | |
9 | 6, 7, 8 | addcomli 10820 | . . 3 ⊢ (2 + 4) = 6 |
10 | 9 | oveq2i 7156 | . 2 ⊢ (4 gcd (2 + 4)) = (4 gcd 6) |
11 | 2z 12002 | . . . . 5 ⊢ 2 ∈ ℤ | |
12 | gcdadd 15862 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 2 ∈ ℤ) → (2 gcd 2) = (2 gcd (2 + 2))) | |
13 | 11, 11, 12 | mp2an 688 | . . . 4 ⊢ (2 gcd 2) = (2 gcd (2 + 2)) |
14 | 2p2e4 11760 | . . . . . 6 ⊢ (2 + 2) = 4 | |
15 | 14 | oveq2i 7156 | . . . . 5 ⊢ (2 gcd (2 + 2)) = (2 gcd 4) |
16 | gcdcom 15850 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → (2 gcd 4) = (4 gcd 2)) | |
17 | 11, 3, 16 | mp2an 688 | . . . . 5 ⊢ (2 gcd 4) = (4 gcd 2) |
18 | 15, 17 | eqtri 2841 | . . . 4 ⊢ (2 gcd (2 + 2)) = (4 gcd 2) |
19 | 13, 18 | eqtri 2841 | . . 3 ⊢ (2 gcd 2) = (4 gcd 2) |
20 | gcdid 15863 | . . . . 5 ⊢ (2 ∈ ℤ → (2 gcd 2) = (abs‘2)) | |
21 | 11, 20 | ax-mp 5 | . . . 4 ⊢ (2 gcd 2) = (abs‘2) |
22 | 2re 11699 | . . . . 5 ⊢ 2 ∈ ℝ | |
23 | 0le2 11727 | . . . . 5 ⊢ 0 ≤ 2 | |
24 | absid 14644 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) | |
25 | 22, 23, 24 | mp2an 688 | . . . 4 ⊢ (abs‘2) = 2 |
26 | 21, 25 | eqtri 2841 | . . 3 ⊢ (2 gcd 2) = 2 |
27 | gcdadd 15862 | . . . 4 ⊢ ((4 ∈ ℤ ∧ 2 ∈ ℤ) → (4 gcd 2) = (4 gcd (2 + 4))) | |
28 | 3, 11, 27 | mp2an 688 | . . 3 ⊢ (4 gcd 2) = (4 gcd (2 + 4)) |
29 | 19, 26, 28 | 3eqtr3ri 2850 | . 2 ⊢ (4 gcd (2 + 4)) = 2 |
30 | 5, 10, 29 | 3eqtr2i 2847 | 1 ⊢ (6 gcd 4) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 + caddc 10528 ≤ cle 10664 2c2 11680 4c4 11682 6c6 11684 ℤcz 11969 abscabs 14581 gcd cgcd 15831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 |
This theorem is referenced by: 6lcm4e12 15948 |
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