![]() |
Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 12gcd5e1 | Structured version Visualization version GIF version |
Description: The gcd of 12 and 5 is 1. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
12gcd5e1 | ⊢ (;12 gcd 5) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lt5 12339 | . . . . . 6 ⊢ 2 < 5 | |
2 | 1 | olci 865 | . . . . 5 ⊢ (5 < 2 ∨ 2 < 5) |
3 | 5re 12247 | . . . . . 6 ⊢ 5 ∈ ℝ | |
4 | 2re 12234 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | lttri2 11244 | . . . . . 6 ⊢ ((5 ∈ ℝ ∧ 2 ∈ ℝ) → (5 ≠ 2 ↔ (5 < 2 ∨ 2 < 5))) | |
6 | 3, 4, 5 | mp2an 691 | . . . . 5 ⊢ (5 ≠ 2 ↔ (5 < 2 ∨ 2 < 5)) |
7 | 2, 6 | mpbir 230 | . . . 4 ⊢ 5 ≠ 2 |
8 | 5prm 16988 | . . . . 5 ⊢ 5 ∈ ℙ | |
9 | 2prm 16575 | . . . . 5 ⊢ 2 ∈ ℙ | |
10 | prmrp 16595 | . . . . 5 ⊢ ((5 ∈ ℙ ∧ 2 ∈ ℙ) → ((5 gcd 2) = 1 ↔ 5 ≠ 2)) | |
11 | 8, 9, 10 | mp2an 691 | . . . 4 ⊢ ((5 gcd 2) = 1 ↔ 5 ≠ 2) |
12 | 7, 11 | mpbir 230 | . . 3 ⊢ (5 gcd 2) = 1 |
13 | 5nn 12246 | . . . . 5 ⊢ 5 ∈ ℕ | |
14 | 2nn 12233 | . . . . 5 ⊢ 2 ∈ ℕ | |
15 | 14 | nnzi 12534 | . . . . 5 ⊢ 2 ∈ ℤ |
16 | 13, 14, 15 | gcdaddmzz2nncomi 40482 | . . . 4 ⊢ (5 gcd 2) = (5 gcd ((2 · 5) + 2)) |
17 | 13, 14 | mulcomnni 40474 | . . . . . . . 8 ⊢ (5 · 2) = (2 · 5) |
18 | 5t2e10 12725 | . . . . . . . 8 ⊢ (5 · 2) = ;10 | |
19 | 17, 18 | eqtr3i 2767 | . . . . . . 7 ⊢ (2 · 5) = ;10 |
20 | 19 | oveq1i 7372 | . . . . . 6 ⊢ ((2 · 5) + 2) = (;10 + 2) |
21 | 1nn0 12436 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
22 | 0nn0 12435 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
23 | 14 | nnnn0i 12428 | . . . . . . 7 ⊢ 2 ∈ ℕ0 |
24 | eqid 2737 | . . . . . . 7 ⊢ ;10 = ;10 | |
25 | 23 | dec0h 12647 | . . . . . . 7 ⊢ 2 = ;02 |
26 | 1p0e1 12284 | . . . . . . 7 ⊢ (1 + 0) = 1 | |
27 | 2cn 12235 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
28 | 27 | addid2i 11350 | . . . . . . 7 ⊢ (0 + 2) = 2 |
29 | 21, 22, 22, 23, 24, 25, 26, 28 | decadd 12679 | . . . . . 6 ⊢ (;10 + 2) = ;12 |
30 | 20, 29 | eqtri 2765 | . . . . 5 ⊢ ((2 · 5) + 2) = ;12 |
31 | 30 | oveq2i 7373 | . . . 4 ⊢ (5 gcd ((2 · 5) + 2)) = (5 gcd ;12) |
32 | 16, 31 | eqtri 2765 | . . 3 ⊢ (5 gcd 2) = (5 gcd ;12) |
33 | 12, 32 | eqtr3i 2767 | . 2 ⊢ 1 = (5 gcd ;12) |
34 | 21, 14 | decnncl 12645 | . . 3 ⊢ ;12 ∈ ℕ |
35 | 13, 34 | gcdcomnni 40475 | . 2 ⊢ (5 gcd ;12) = (;12 gcd 5) |
36 | 33, 35 | eqtr2i 2766 | 1 ⊢ (;12 gcd 5) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 class class class wbr 5110 (class class class)co 7362 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 · cmul 11063 < clt 11196 2c2 12215 5c5 12218 ;cdc 12625 gcd cgcd 16381 ℙcprime 16554 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-fz 13432 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-gcd 16382 df-prm 16555 |
This theorem is referenced by: 12lcm5e60 40494 |
Copyright terms: Public domain | W3C validator |