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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 12339 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 12624 | . . 3 ⊢ 6 ∈ ℤ |
3 | 4z 12634 | . . 3 ⊢ 4 ∈ ℤ | |
4 | gcdcom 16495 | . . 3 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) = (4 gcd 6)) | |
5 | 2, 3, 4 | mp2an 690 | . 2 ⊢ (6 gcd 4) = (4 gcd 6) |
6 | 4cn 12335 | . . . 4 ⊢ 4 ∈ ℂ | |
7 | 2cn 12325 | . . . 4 ⊢ 2 ∈ ℂ | |
8 | 4p2e6 12403 | . . . 4 ⊢ (4 + 2) = 6 | |
9 | 6, 7, 8 | addcomli 11444 | . . 3 ⊢ (2 + 4) = 6 |
10 | 9 | oveq2i 7437 | . 2 ⊢ (4 gcd (2 + 4)) = (4 gcd 6) |
11 | 2z 12632 | . . . . 5 ⊢ 2 ∈ ℤ | |
12 | gcdadd 16508 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 2 ∈ ℤ) → (2 gcd 2) = (2 gcd (2 + 2))) | |
13 | 11, 11, 12 | mp2an 690 | . . . 4 ⊢ (2 gcd 2) = (2 gcd (2 + 2)) |
14 | 2p2e4 12385 | . . . . . 6 ⊢ (2 + 2) = 4 | |
15 | 14 | oveq2i 7437 | . . . . 5 ⊢ (2 gcd (2 + 2)) = (2 gcd 4) |
16 | gcdcom 16495 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → (2 gcd 4) = (4 gcd 2)) | |
17 | 11, 3, 16 | mp2an 690 | . . . . 5 ⊢ (2 gcd 4) = (4 gcd 2) |
18 | 15, 17 | eqtri 2756 | . . . 4 ⊢ (2 gcd (2 + 2)) = (4 gcd 2) |
19 | 13, 18 | eqtri 2756 | . . 3 ⊢ (2 gcd 2) = (4 gcd 2) |
20 | gcdid 16509 | . . . . 5 ⊢ (2 ∈ ℤ → (2 gcd 2) = (abs‘2)) | |
21 | 11, 20 | ax-mp 5 | . . . 4 ⊢ (2 gcd 2) = (abs‘2) |
22 | 2re 12324 | . . . . 5 ⊢ 2 ∈ ℝ | |
23 | 0le2 12352 | . . . . 5 ⊢ 0 ≤ 2 | |
24 | absid 15283 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) | |
25 | 22, 23, 24 | mp2an 690 | . . . 4 ⊢ (abs‘2) = 2 |
26 | 21, 25 | eqtri 2756 | . . 3 ⊢ (2 gcd 2) = 2 |
27 | gcdadd 16508 | . . . 4 ⊢ ((4 ∈ ℤ ∧ 2 ∈ ℤ) → (4 gcd 2) = (4 gcd (2 + 4))) | |
28 | 3, 11, 27 | mp2an 690 | . . 3 ⊢ (4 gcd 2) = (4 gcd (2 + 4)) |
29 | 19, 26, 28 | 3eqtr3ri 2765 | . 2 ⊢ (4 gcd (2 + 4)) = 2 |
30 | 5, 10, 29 | 3eqtr2i 2762 | 1 ⊢ (6 gcd 4) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 0cc0 11146 + caddc 11149 ≤ cle 11287 2c2 12305 4c4 12307 6c6 12309 ℤcz 12596 abscabs 15221 gcd cgcd 16476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-dvds 16239 df-gcd 16477 |
This theorem is referenced by: 6lcm4e12 16594 |
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