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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 12332 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 12617 | . . 3 ⊢ 6 ∈ ℤ |
3 | 4z 12627 | . . 3 ⊢ 4 ∈ ℤ | |
4 | gcdcom 16488 | . . 3 ⊢ ((6 ∈ ℤ ∧ 4 ∈ ℤ) → (6 gcd 4) = (4 gcd 6)) | |
5 | 2, 3, 4 | mp2an 691 | . 2 ⊢ (6 gcd 4) = (4 gcd 6) |
6 | 4cn 12328 | . . . 4 ⊢ 4 ∈ ℂ | |
7 | 2cn 12318 | . . . 4 ⊢ 2 ∈ ℂ | |
8 | 4p2e6 12396 | . . . 4 ⊢ (4 + 2) = 6 | |
9 | 6, 7, 8 | addcomli 11437 | . . 3 ⊢ (2 + 4) = 6 |
10 | 9 | oveq2i 7431 | . 2 ⊢ (4 gcd (2 + 4)) = (4 gcd 6) |
11 | 2z 12625 | . . . . 5 ⊢ 2 ∈ ℤ | |
12 | gcdadd 16501 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 2 ∈ ℤ) → (2 gcd 2) = (2 gcd (2 + 2))) | |
13 | 11, 11, 12 | mp2an 691 | . . . 4 ⊢ (2 gcd 2) = (2 gcd (2 + 2)) |
14 | 2p2e4 12378 | . . . . . 6 ⊢ (2 + 2) = 4 | |
15 | 14 | oveq2i 7431 | . . . . 5 ⊢ (2 gcd (2 + 2)) = (2 gcd 4) |
16 | gcdcom 16488 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → (2 gcd 4) = (4 gcd 2)) | |
17 | 11, 3, 16 | mp2an 691 | . . . . 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 16502 | . . . . 5 ⊢ (2 ∈ ℤ → (2 gcd 2) = (abs‘2)) | |
21 | 11, 20 | ax-mp 5 | . . . 4 ⊢ (2 gcd 2) = (abs‘2) |
22 | 2re 12317 | . . . . 5 ⊢ 2 ∈ ℝ | |
23 | 0le2 12345 | . . . . 5 ⊢ 0 ≤ 2 | |
24 | absid 15276 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) | |
25 | 22, 23, 24 | mp2an 691 | . . . 4 ⊢ (abs‘2) = 2 |
26 | 21, 25 | eqtri 2756 | . . 3 ⊢ (2 gcd 2) = 2 |
27 | gcdadd 16501 | . . . 4 ⊢ ((4 ∈ ℤ ∧ 2 ∈ ℤ) → (4 gcd 2) = (4 gcd (2 + 4))) | |
28 | 3, 11, 27 | mp2an 691 | . . 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 1534 ∈ wcel 2099 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 ℝcr 11138 0cc0 11139 + caddc 11142 ≤ cle 11280 2c2 12298 4c4 12300 6c6 12302 ℤcz 12589 abscabs 15214 gcd cgcd 16469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-dvds 16232 df-gcd 16470 |
This theorem is referenced by: 6lcm4e12 16587 |
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