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Mirrors > Home > MPE Home > Th. List > Mathboxes > 60gcd7e1 | Structured version Visualization version GIF version |
Description: The gcd of 60 and 7 is 1. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
60gcd7e1 | ⊢ (;60 gcd 7) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7nn 12203 | . . 3 ⊢ 7 ∈ ℕ | |
2 | 6nn 12200 | . . . 4 ⊢ 6 ∈ ℕ | |
3 | 2 | decnncl2 12600 | . . 3 ⊢ ;60 ∈ ℕ |
4 | 1, 3 | gcdcomnni 40378 | . 2 ⊢ (7 gcd ;60) = (;60 gcd 7) |
5 | 1nn0 12387 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1nn 12122 | . . . . . 6 ⊢ 1 ∈ ℕ | |
7 | 5, 6 | decnncl 12596 | . . . . 5 ⊢ ;11 ∈ ℕ |
8 | 1 | nnzi 12485 | . . . . 5 ⊢ 7 ∈ ℤ |
9 | 1, 7, 8 | gcdaddmzz2nni 40384 | . . . 4 ⊢ (7 gcd ;11) = (7 gcd (;11 + (7 · 7))) |
10 | 7t7e49 12690 | . . . . . . 7 ⊢ (7 · 7) = ;49 | |
11 | 10 | oveq2i 7362 | . . . . . 6 ⊢ (;11 + (7 · 7)) = (;11 + ;49) |
12 | 4nn0 12390 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
13 | 9nn0 12395 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
14 | eqid 2736 | . . . . . . 7 ⊢ ;11 = ;11 | |
15 | eqid 2736 | . . . . . . 7 ⊢ ;49 = ;49 | |
16 | 4cn 12196 | . . . . . . . . . 10 ⊢ 4 ∈ ℂ | |
17 | ax-1cn 11067 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
18 | 4p1e5 12257 | . . . . . . . . . 10 ⊢ (4 + 1) = 5 | |
19 | 16, 17, 18 | addcomli 11305 | . . . . . . . . 9 ⊢ (1 + 4) = 5 |
20 | 19 | oveq1i 7361 | . . . . . . . 8 ⊢ ((1 + 4) + 1) = (5 + 1) |
21 | 5p1e6 12258 | . . . . . . . 8 ⊢ (5 + 1) = 6 | |
22 | 20, 21 | eqtri 2764 | . . . . . . 7 ⊢ ((1 + 4) + 1) = 6 |
23 | 9cn 12211 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
24 | 9p1e10 12578 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
25 | 23, 17, 24 | addcomli 11305 | . . . . . . 7 ⊢ (1 + 9) = ;10 |
26 | 5, 5, 12, 13, 14, 15, 22, 25 | decaddc2 12632 | . . . . . 6 ⊢ (;11 + ;49) = ;60 |
27 | 11, 26 | eqtri 2764 | . . . . 5 ⊢ (;11 + (7 · 7)) = ;60 |
28 | 27 | oveq2i 7362 | . . . 4 ⊢ (7 gcd (;11 + (7 · 7))) = (7 gcd ;60) |
29 | 9, 28 | eqtri 2764 | . . 3 ⊢ (7 gcd ;11) = (7 gcd ;60) |
30 | 7re 12204 | . . . . . 6 ⊢ 7 ∈ ℝ | |
31 | 1 | nnnn0i 12379 | . . . . . . . 8 ⊢ 7 ∈ ℕ0 |
32 | 31 | dec0h 12598 | . . . . . . 7 ⊢ 7 = ;07 |
33 | 0nn0 12386 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
34 | 7lt9 12311 | . . . . . . . . 9 ⊢ 7 < 9 | |
35 | 9re 12210 | . . . . . . . . . . 11 ⊢ 9 ∈ ℝ | |
36 | 30, 35 | pm3.2i 471 | . . . . . . . . . 10 ⊢ (7 ∈ ℝ ∧ 9 ∈ ℝ) |
37 | ltle 11201 | . . . . . . . . . 10 ⊢ ((7 ∈ ℝ ∧ 9 ∈ ℝ) → (7 < 9 → 7 ≤ 9)) | |
38 | 36, 37 | ax-mp 5 | . . . . . . . . 9 ⊢ (7 < 9 → 7 ≤ 9) |
39 | 34, 38 | ax-mp 5 | . . . . . . . 8 ⊢ 7 ≤ 9 |
40 | 0lt1 11635 | . . . . . . . 8 ⊢ 0 < 1 | |
41 | 33, 5, 31, 5, 39, 40 | declth 12606 | . . . . . . 7 ⊢ ;07 < ;11 |
42 | 32, 41 | eqbrtri 5124 | . . . . . 6 ⊢ 7 < ;11 |
43 | ltne 11210 | . . . . . 6 ⊢ ((7 ∈ ℝ ∧ 7 < ;11) → ;11 ≠ 7) | |
44 | 30, 42, 43 | mp2an 690 | . . . . 5 ⊢ ;11 ≠ 7 |
45 | necom 2995 | . . . . 5 ⊢ (7 ≠ ;11 ↔ ;11 ≠ 7) | |
46 | 44, 45 | mpbir 230 | . . . 4 ⊢ 7 ≠ ;11 |
47 | 7prm 16937 | . . . . 5 ⊢ 7 ∈ ℙ | |
48 | 11prm 16941 | . . . . 5 ⊢ ;11 ∈ ℙ | |
49 | prmrp 16542 | . . . . 5 ⊢ ((7 ∈ ℙ ∧ ;11 ∈ ℙ) → ((7 gcd ;11) = 1 ↔ 7 ≠ ;11)) | |
50 | 47, 48, 49 | mp2an 690 | . . . 4 ⊢ ((7 gcd ;11) = 1 ↔ 7 ≠ ;11) |
51 | 46, 50 | mpbir 230 | . . 3 ⊢ (7 gcd ;11) = 1 |
52 | 29, 51 | eqtr3i 2766 | . 2 ⊢ (7 gcd ;60) = 1 |
53 | 4, 52 | eqtr3i 2766 | 1 ⊢ (;60 gcd 7) = 1 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 class class class wbr 5103 (class class class)co 7351 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 < clt 11147 ≤ cle 11148 4c4 12168 5c5 12169 6c6 12170 7c7 12171 9c9 12173 ;cdc 12576 gcd cgcd 16328 ℙcprime 16501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-rp 12870 df-fz 13379 df-seq 13861 df-exp 13922 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-dvds 16091 df-gcd 16329 df-prm 16502 |
This theorem is referenced by: 60lcm7e420 40399 |
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