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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 11716 | . . 3 ⊢ 7 ∈ ℕ | |
2 | 6nn 11713 | . . . 4 ⊢ 6 ∈ ℕ | |
3 | 2 | decnncl2 12109 | . . 3 ⊢ ;60 ∈ ℕ |
4 | 1, 3 | gcdcomnni 39134 | . 2 ⊢ (7 gcd ;60) = (;60 gcd 7) |
5 | 1nn0 11900 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1nn 11635 | . . . . . 6 ⊢ 1 ∈ ℕ | |
7 | 5, 6 | decnncl 12105 | . . . . 5 ⊢ ;11 ∈ ℕ |
8 | 1 | nnzi 11993 | . . . . 5 ⊢ 7 ∈ ℤ |
9 | 1, 7, 8 | gcdaddmzz2nni 39138 | . . . 4 ⊢ (7 gcd ;11) = (7 gcd (;11 + (7 · 7))) |
10 | 7t7e49 12199 | . . . . . . 7 ⊢ (7 · 7) = ;49 | |
11 | 10 | oveq2i 7153 | . . . . . 6 ⊢ (;11 + (7 · 7)) = (;11 + ;49) |
12 | 4nn0 11903 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
13 | 9nn0 11908 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
14 | eqid 2821 | . . . . . . 7 ⊢ ;11 = ;11 | |
15 | eqid 2821 | . . . . . . 7 ⊢ ;49 = ;49 | |
16 | 4cn 11709 | . . . . . . . . . 10 ⊢ 4 ∈ ℂ | |
17 | ax-1cn 10581 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
18 | 4p1e5 11770 | . . . . . . . . . 10 ⊢ (4 + 1) = 5 | |
19 | 16, 17, 18 | addcomli 10818 | . . . . . . . . 9 ⊢ (1 + 4) = 5 |
20 | 19 | oveq1i 7152 | . . . . . . . 8 ⊢ ((1 + 4) + 1) = (5 + 1) |
21 | 5p1e6 11771 | . . . . . . . 8 ⊢ (5 + 1) = 6 | |
22 | 20, 21 | eqtri 2844 | . . . . . . 7 ⊢ ((1 + 4) + 1) = 6 |
23 | 9cn 11724 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
24 | 9p1e10 12087 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
25 | 23, 17, 24 | addcomli 10818 | . . . . . . 7 ⊢ (1 + 9) = ;10 |
26 | 5, 5, 12, 13, 14, 15, 22, 25 | decaddc2 12141 | . . . . . 6 ⊢ (;11 + ;49) = ;60 |
27 | 11, 26 | eqtri 2844 | . . . . 5 ⊢ (;11 + (7 · 7)) = ;60 |
28 | 27 | oveq2i 7153 | . . . 4 ⊢ (7 gcd (;11 + (7 · 7))) = (7 gcd ;60) |
29 | 9, 28 | eqtri 2844 | . . 3 ⊢ (7 gcd ;11) = (7 gcd ;60) |
30 | 7re 11717 | . . . . . 6 ⊢ 7 ∈ ℝ | |
31 | 1 | nnnn0i 11892 | . . . . . . . 8 ⊢ 7 ∈ ℕ0 |
32 | 31 | dec0h 12107 | . . . . . . 7 ⊢ 7 = ;07 |
33 | 0nn0 11899 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
34 | 7lt9 11824 | . . . . . . . . 9 ⊢ 7 < 9 | |
35 | 9re 11723 | . . . . . . . . . . 11 ⊢ 9 ∈ ℝ | |
36 | 30, 35 | pm3.2i 473 | . . . . . . . . . 10 ⊢ (7 ∈ ℝ ∧ 9 ∈ ℝ) |
37 | ltle 10715 | . . . . . . . . . 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 11148 | . . . . . . . 8 ⊢ 0 < 1 | |
41 | 33, 5, 31, 5, 39, 40 | declth 12115 | . . . . . . 7 ⊢ ;07 < ;11 |
42 | 32, 41 | eqbrtri 5073 | . . . . . 6 ⊢ 7 < ;11 |
43 | ltne 10723 | . . . . . 6 ⊢ ((7 ∈ ℝ ∧ 7 < ;11) → ;11 ≠ 7) | |
44 | 30, 42, 43 | mp2an 690 | . . . . 5 ⊢ ;11 ≠ 7 |
45 | necom 3069 | . . . . 5 ⊢ (7 ≠ ;11 ↔ ;11 ≠ 7) | |
46 | 44, 45 | mpbir 233 | . . . 4 ⊢ 7 ≠ ;11 |
47 | 7prm 16427 | . . . . 5 ⊢ 7 ∈ ℙ | |
48 | 11prm 16431 | . . . . 5 ⊢ ;11 ∈ ℙ | |
49 | prmrp 16039 | . . . . 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 233 | . . 3 ⊢ (7 gcd ;11) = 1 |
52 | 29, 51 | eqtr3i 2846 | . 2 ⊢ (7 gcd ;60) = 1 |
53 | 4, 52 | eqtr3i 2846 | 1 ⊢ (;60 gcd 7) = 1 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5052 (class class class)co 7142 ℝcr 10522 0cc0 10523 1c1 10524 + caddc 10526 · cmul 10528 < clt 10661 ≤ cle 10662 4c4 11681 5c5 11682 6c6 11683 7c7 11684 9c9 11686 ;cdc 12085 gcd cgcd 15826 ℙcprime 15998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-rp 12377 df-fz 12883 df-seq 13360 df-exp 13420 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-dvds 15593 df-gcd 15827 df-prm 15999 |
This theorem is referenced by: 60lcm7e420 39148 |
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