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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 12252 | . . 3 ⊢ 7 ∈ ℕ | |
2 | 6nn 12249 | . . . 4 ⊢ 6 ∈ ℕ | |
3 | 2 | decnncl2 12649 | . . 3 ⊢ ;60 ∈ ℕ |
4 | 1, 3 | gcdcomnni 40475 | . 2 ⊢ (7 gcd ;60) = (;60 gcd 7) |
5 | 1nn0 12436 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1nn 12171 | . . . . . 6 ⊢ 1 ∈ ℕ | |
7 | 5, 6 | decnncl 12645 | . . . . 5 ⊢ ;11 ∈ ℕ |
8 | 1 | nnzi 12534 | . . . . 5 ⊢ 7 ∈ ℤ |
9 | 1, 7, 8 | gcdaddmzz2nni 40481 | . . . 4 ⊢ (7 gcd ;11) = (7 gcd (;11 + (7 · 7))) |
10 | 7t7e49 12739 | . . . . . . 7 ⊢ (7 · 7) = ;49 | |
11 | 10 | oveq2i 7373 | . . . . . 6 ⊢ (;11 + (7 · 7)) = (;11 + ;49) |
12 | 4nn0 12439 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
13 | 9nn0 12444 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
14 | eqid 2737 | . . . . . . 7 ⊢ ;11 = ;11 | |
15 | eqid 2737 | . . . . . . 7 ⊢ ;49 = ;49 | |
16 | 4cn 12245 | . . . . . . . . . 10 ⊢ 4 ∈ ℂ | |
17 | ax-1cn 11116 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
18 | 4p1e5 12306 | . . . . . . . . . 10 ⊢ (4 + 1) = 5 | |
19 | 16, 17, 18 | addcomli 11354 | . . . . . . . . 9 ⊢ (1 + 4) = 5 |
20 | 19 | oveq1i 7372 | . . . . . . . 8 ⊢ ((1 + 4) + 1) = (5 + 1) |
21 | 5p1e6 12307 | . . . . . . . 8 ⊢ (5 + 1) = 6 | |
22 | 20, 21 | eqtri 2765 | . . . . . . 7 ⊢ ((1 + 4) + 1) = 6 |
23 | 9cn 12260 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
24 | 9p1e10 12627 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
25 | 23, 17, 24 | addcomli 11354 | . . . . . . 7 ⊢ (1 + 9) = ;10 |
26 | 5, 5, 12, 13, 14, 15, 22, 25 | decaddc2 12681 | . . . . . 6 ⊢ (;11 + ;49) = ;60 |
27 | 11, 26 | eqtri 2765 | . . . . 5 ⊢ (;11 + (7 · 7)) = ;60 |
28 | 27 | oveq2i 7373 | . . . 4 ⊢ (7 gcd (;11 + (7 · 7))) = (7 gcd ;60) |
29 | 9, 28 | eqtri 2765 | . . 3 ⊢ (7 gcd ;11) = (7 gcd ;60) |
30 | 7re 12253 | . . . . . 6 ⊢ 7 ∈ ℝ | |
31 | 1 | nnnn0i 12428 | . . . . . . . 8 ⊢ 7 ∈ ℕ0 |
32 | 31 | dec0h 12647 | . . . . . . 7 ⊢ 7 = ;07 |
33 | 0nn0 12435 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
34 | 7lt9 12360 | . . . . . . . . 9 ⊢ 7 < 9 | |
35 | 9re 12259 | . . . . . . . . . . 11 ⊢ 9 ∈ ℝ | |
36 | 30, 35 | pm3.2i 472 | . . . . . . . . . 10 ⊢ (7 ∈ ℝ ∧ 9 ∈ ℝ) |
37 | ltle 11250 | . . . . . . . . . 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 11684 | . . . . . . . 8 ⊢ 0 < 1 | |
41 | 33, 5, 31, 5, 39, 40 | declth 12655 | . . . . . . 7 ⊢ ;07 < ;11 |
42 | 32, 41 | eqbrtri 5131 | . . . . . 6 ⊢ 7 < ;11 |
43 | ltne 11259 | . . . . . 6 ⊢ ((7 ∈ ℝ ∧ 7 < ;11) → ;11 ≠ 7) | |
44 | 30, 42, 43 | mp2an 691 | . . . . 5 ⊢ ;11 ≠ 7 |
45 | necom 2998 | . . . . 5 ⊢ (7 ≠ ;11 ↔ ;11 ≠ 7) | |
46 | 44, 45 | mpbir 230 | . . . 4 ⊢ 7 ≠ ;11 |
47 | 7prm 16990 | . . . . 5 ⊢ 7 ∈ ℙ | |
48 | 11prm 16994 | . . . . 5 ⊢ ;11 ∈ ℙ | |
49 | prmrp 16595 | . . . . 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 230 | . . 3 ⊢ (7 gcd ;11) = 1 |
52 | 29, 51 | eqtr3i 2767 | . 2 ⊢ (7 gcd ;60) = 1 |
53 | 4, 52 | eqtr3i 2767 | 1 ⊢ (;60 gcd 7) = 1 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = 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 ≤ cle 11197 4c4 12217 5c5 12218 6c6 12219 7c7 12220 9c9 12222 ;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: 60lcm7e420 40496 |
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