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| 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 12445 | . . . . . 6 ⊢ 2 < 5 | |
| 2 | 1 | olci 867 | . . . . 5 ⊢ (5 < 2 ∨ 2 < 5) |
| 3 | 5re 12353 | . . . . . 6 ⊢ 5 ∈ ℝ | |
| 4 | 2re 12340 | . . . . . 6 ⊢ 2 ∈ ℝ | |
| 5 | lttri2 11343 | . . . . . 6 ⊢ ((5 ∈ ℝ ∧ 2 ∈ ℝ) → (5 ≠ 2 ↔ (5 < 2 ∨ 2 < 5))) | |
| 6 | 3, 4, 5 | mp2an 692 | . . . . 5 ⊢ (5 ≠ 2 ↔ (5 < 2 ∨ 2 < 5)) |
| 7 | 2, 6 | mpbir 231 | . . . 4 ⊢ 5 ≠ 2 |
| 8 | 5prm 17146 | . . . . 5 ⊢ 5 ∈ ℙ | |
| 9 | 2prm 16729 | . . . . 5 ⊢ 2 ∈ ℙ | |
| 10 | prmrp 16749 | . . . . 5 ⊢ ((5 ∈ ℙ ∧ 2 ∈ ℙ) → ((5 gcd 2) = 1 ↔ 5 ≠ 2)) | |
| 11 | 8, 9, 10 | mp2an 692 | . . . 4 ⊢ ((5 gcd 2) = 1 ↔ 5 ≠ 2) |
| 12 | 7, 11 | mpbir 231 | . . 3 ⊢ (5 gcd 2) = 1 |
| 13 | 5nn 12352 | . . . . 5 ⊢ 5 ∈ ℕ | |
| 14 | 2nn 12339 | . . . . 5 ⊢ 2 ∈ ℕ | |
| 15 | 14 | nnzi 12641 | . . . . 5 ⊢ 2 ∈ ℤ |
| 16 | 13, 14, 15 | gcdaddmzz2nncomi 41996 | . . . 4 ⊢ (5 gcd 2) = (5 gcd ((2 · 5) + 2)) |
| 17 | 13, 14 | mulcomnni 41988 | . . . . . . . 8 ⊢ (5 · 2) = (2 · 5) |
| 18 | 5t2e10 12833 | . . . . . . . 8 ⊢ (5 · 2) = ;10 | |
| 19 | 17, 18 | eqtr3i 2767 | . . . . . . 7 ⊢ (2 · 5) = ;10 |
| 20 | 19 | oveq1i 7441 | . . . . . 6 ⊢ ((2 · 5) + 2) = (;10 + 2) |
| 21 | 1nn0 12542 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 22 | 0nn0 12541 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
| 23 | 14 | nnnn0i 12534 | . . . . . . 7 ⊢ 2 ∈ ℕ0 |
| 24 | eqid 2737 | . . . . . . 7 ⊢ ;10 = ;10 | |
| 25 | 23 | dec0h 12755 | . . . . . . 7 ⊢ 2 = ;02 |
| 26 | 1p0e1 12390 | . . . . . . 7 ⊢ (1 + 0) = 1 | |
| 27 | 2cn 12341 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
| 28 | 27 | addlidi 11449 | . . . . . . 7 ⊢ (0 + 2) = 2 |
| 29 | 21, 22, 22, 23, 24, 25, 26, 28 | decadd 12787 | . . . . . 6 ⊢ (;10 + 2) = ;12 |
| 30 | 20, 29 | eqtri 2765 | . . . . 5 ⊢ ((2 · 5) + 2) = ;12 |
| 31 | 30 | oveq2i 7442 | . . . 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 12753 | . . 3 ⊢ ;12 ∈ ℕ |
| 35 | 13, 34 | gcdcomnni 41989 | . 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 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 2c2 12321 5c5 12324 ;cdc 12733 gcd cgcd 16531 ℙcprime 16708 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 |
| This theorem is referenced by: 12lcm5e60 42009 |
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