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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 12247 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 12532 | . . 3 ⊢ 6 ∈ ℤ |
3 | 4z 12542 | . . 3 ⊢ 4 ∈ ℤ | |
4 | gcdcom 16398 | . . 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 12243 | . . . 4 ⊢ 4 ∈ ℂ | |
7 | 2cn 12233 | . . . 4 ⊢ 2 ∈ ℂ | |
8 | 4p2e6 12311 | . . . 4 ⊢ (4 + 2) = 6 | |
9 | 6, 7, 8 | addcomli 11352 | . . 3 ⊢ (2 + 4) = 6 |
10 | 9 | oveq2i 7369 | . 2 ⊢ (4 gcd (2 + 4)) = (4 gcd 6) |
11 | 2z 12540 | . . . . 5 ⊢ 2 ∈ ℤ | |
12 | gcdadd 16411 | . . . . 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 12293 | . . . . . 6 ⊢ (2 + 2) = 4 | |
15 | 14 | oveq2i 7369 | . . . . 5 ⊢ (2 gcd (2 + 2)) = (2 gcd 4) |
16 | gcdcom 16398 | . . . . . 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 2761 | . . . 4 ⊢ (2 gcd (2 + 2)) = (4 gcd 2) |
19 | 13, 18 | eqtri 2761 | . . 3 ⊢ (2 gcd 2) = (4 gcd 2) |
20 | gcdid 16412 | . . . . 5 ⊢ (2 ∈ ℤ → (2 gcd 2) = (abs‘2)) | |
21 | 11, 20 | ax-mp 5 | . . . 4 ⊢ (2 gcd 2) = (abs‘2) |
22 | 2re 12232 | . . . . 5 ⊢ 2 ∈ ℝ | |
23 | 0le2 12260 | . . . . 5 ⊢ 0 ≤ 2 | |
24 | absid 15187 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) | |
25 | 22, 23, 24 | mp2an 691 | . . . 4 ⊢ (abs‘2) = 2 |
26 | 21, 25 | eqtri 2761 | . . 3 ⊢ (2 gcd 2) = 2 |
27 | gcdadd 16411 | . . . 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 2770 | . 2 ⊢ (4 gcd (2 + 4)) = 2 |
30 | 5, 10, 29 | 3eqtr2i 2767 | 1 ⊢ (6 gcd 4) = 2 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 ℝcr 11055 0cc0 11056 + caddc 11059 ≤ cle 11195 2c2 12213 4c4 12215 6c6 12217 ℤcz 12504 abscabs 15125 gcd cgcd 16379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-dvds 16142 df-gcd 16380 |
This theorem is referenced by: 6lcm4e12 16497 |
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