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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 11717 | . . 3 ⊢ 7 ∈ ℕ | |
2 | 6nn 11714 | . . . 4 ⊢ 6 ∈ ℕ | |
3 | 2 | decnncl2 12110 | . . 3 ⊢ ;60 ∈ ℕ |
4 | 1, 3 | gcdcomnni 39276 | . 2 ⊢ (7 gcd ;60) = (;60 gcd 7) |
5 | 1nn0 11901 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1nn 11636 | . . . . . 6 ⊢ 1 ∈ ℕ | |
7 | 5, 6 | decnncl 12106 | . . . . 5 ⊢ ;11 ∈ ℕ |
8 | 1 | nnzi 11994 | . . . . 5 ⊢ 7 ∈ ℤ |
9 | 1, 7, 8 | gcdaddmzz2nni 39282 | . . . 4 ⊢ (7 gcd ;11) = (7 gcd (;11 + (7 · 7))) |
10 | 7t7e49 12200 | . . . . . . 7 ⊢ (7 · 7) = ;49 | |
11 | 10 | oveq2i 7146 | . . . . . 6 ⊢ (;11 + (7 · 7)) = (;11 + ;49) |
12 | 4nn0 11904 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
13 | 9nn0 11909 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
14 | eqid 2798 | . . . . . . 7 ⊢ ;11 = ;11 | |
15 | eqid 2798 | . . . . . . 7 ⊢ ;49 = ;49 | |
16 | 4cn 11710 | . . . . . . . . . 10 ⊢ 4 ∈ ℂ | |
17 | ax-1cn 10584 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
18 | 4p1e5 11771 | . . . . . . . . . 10 ⊢ (4 + 1) = 5 | |
19 | 16, 17, 18 | addcomli 10821 | . . . . . . . . 9 ⊢ (1 + 4) = 5 |
20 | 19 | oveq1i 7145 | . . . . . . . 8 ⊢ ((1 + 4) + 1) = (5 + 1) |
21 | 5p1e6 11772 | . . . . . . . 8 ⊢ (5 + 1) = 6 | |
22 | 20, 21 | eqtri 2821 | . . . . . . 7 ⊢ ((1 + 4) + 1) = 6 |
23 | 9cn 11725 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
24 | 9p1e10 12088 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
25 | 23, 17, 24 | addcomli 10821 | . . . . . . 7 ⊢ (1 + 9) = ;10 |
26 | 5, 5, 12, 13, 14, 15, 22, 25 | decaddc2 12142 | . . . . . 6 ⊢ (;11 + ;49) = ;60 |
27 | 11, 26 | eqtri 2821 | . . . . 5 ⊢ (;11 + (7 · 7)) = ;60 |
28 | 27 | oveq2i 7146 | . . . 4 ⊢ (7 gcd (;11 + (7 · 7))) = (7 gcd ;60) |
29 | 9, 28 | eqtri 2821 | . . 3 ⊢ (7 gcd ;11) = (7 gcd ;60) |
30 | 7re 11718 | . . . . . 6 ⊢ 7 ∈ ℝ | |
31 | 1 | nnnn0i 11893 | . . . . . . . 8 ⊢ 7 ∈ ℕ0 |
32 | 31 | dec0h 12108 | . . . . . . 7 ⊢ 7 = ;07 |
33 | 0nn0 11900 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
34 | 7lt9 11825 | . . . . . . . . 9 ⊢ 7 < 9 | |
35 | 9re 11724 | . . . . . . . . . . 11 ⊢ 9 ∈ ℝ | |
36 | 30, 35 | pm3.2i 474 | . . . . . . . . . 10 ⊢ (7 ∈ ℝ ∧ 9 ∈ ℝ) |
37 | ltle 10718 | . . . . . . . . . 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 11151 | . . . . . . . 8 ⊢ 0 < 1 | |
41 | 33, 5, 31, 5, 39, 40 | declth 12116 | . . . . . . 7 ⊢ ;07 < ;11 |
42 | 32, 41 | eqbrtri 5051 | . . . . . 6 ⊢ 7 < ;11 |
43 | ltne 10726 | . . . . . 6 ⊢ ((7 ∈ ℝ ∧ 7 < ;11) → ;11 ≠ 7) | |
44 | 30, 42, 43 | mp2an 691 | . . . . 5 ⊢ ;11 ≠ 7 |
45 | necom 3040 | . . . . 5 ⊢ (7 ≠ ;11 ↔ ;11 ≠ 7) | |
46 | 44, 45 | mpbir 234 | . . . 4 ⊢ 7 ≠ ;11 |
47 | 7prm 16436 | . . . . 5 ⊢ 7 ∈ ℙ | |
48 | 11prm 16440 | . . . . 5 ⊢ ;11 ∈ ℙ | |
49 | prmrp 16046 | . . . . 5 ⊢ ((7 ∈ ℙ ∧ ;11 ∈ ℙ) → ((7 gcd ;11) = 1 ↔ 7 ≠ ;11)) | |
50 | 47, 48, 49 | mp2an 691 | . . . 4 ⊢ ((7 gcd ;11) = 1 ↔ 7 ≠ ;11) |
51 | 46, 50 | mpbir 234 | . . 3 ⊢ (7 gcd ;11) = 1 |
52 | 29, 51 | eqtr3i 2823 | . 2 ⊢ (7 gcd ;60) = 1 |
53 | 4, 52 | eqtr3i 2823 | 1 ⊢ (;60 gcd 7) = 1 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 4c4 11682 5c5 11683 6c6 11684 7c7 11685 9c9 11687 ;cdc 12086 gcd cgcd 15833 ℙcprime 16005 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-prm 16006 |
This theorem is referenced by: 60lcm7e420 39298 |
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