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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 12358 | . . 3 ⊢ 7 ∈ ℕ | |
| 2 | 6nn 12355 | . . . 4 ⊢ 6 ∈ ℕ | |
| 3 | 2 | decnncl2 12757 | . . 3 ⊢ ;60 ∈ ℕ |
| 4 | 1, 3 | gcdcomnni 41989 | . 2 ⊢ (7 gcd ;60) = (;60 gcd 7) |
| 5 | 1nn0 12542 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
| 6 | 1nn 12277 | . . . . . 6 ⊢ 1 ∈ ℕ | |
| 7 | 5, 6 | decnncl 12753 | . . . . 5 ⊢ ;11 ∈ ℕ |
| 8 | 1 | nnzi 12641 | . . . . 5 ⊢ 7 ∈ ℤ |
| 9 | 1, 7, 8 | gcdaddmzz2nni 41995 | . . . 4 ⊢ (7 gcd ;11) = (7 gcd (;11 + (7 · 7))) |
| 10 | 7t7e49 12847 | . . . . . . 7 ⊢ (7 · 7) = ;49 | |
| 11 | 10 | oveq2i 7442 | . . . . . 6 ⊢ (;11 + (7 · 7)) = (;11 + ;49) |
| 12 | 4nn0 12545 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
| 13 | 9nn0 12550 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
| 14 | eqid 2737 | . . . . . . 7 ⊢ ;11 = ;11 | |
| 15 | eqid 2737 | . . . . . . 7 ⊢ ;49 = ;49 | |
| 16 | 4cn 12351 | . . . . . . . . . 10 ⊢ 4 ∈ ℂ | |
| 17 | ax-1cn 11213 | . . . . . . . . . 10 ⊢ 1 ∈ ℂ | |
| 18 | 4p1e5 12412 | . . . . . . . . . 10 ⊢ (4 + 1) = 5 | |
| 19 | 16, 17, 18 | addcomli 11453 | . . . . . . . . 9 ⊢ (1 + 4) = 5 |
| 20 | 19 | oveq1i 7441 | . . . . . . . 8 ⊢ ((1 + 4) + 1) = (5 + 1) |
| 21 | 5p1e6 12413 | . . . . . . . 8 ⊢ (5 + 1) = 6 | |
| 22 | 20, 21 | eqtri 2765 | . . . . . . 7 ⊢ ((1 + 4) + 1) = 6 |
| 23 | 9cn 12366 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
| 24 | 9p1e10 12735 | . . . . . . . 8 ⊢ (9 + 1) = ;10 | |
| 25 | 23, 17, 24 | addcomli 11453 | . . . . . . 7 ⊢ (1 + 9) = ;10 |
| 26 | 5, 5, 12, 13, 14, 15, 22, 25 | decaddc2 12789 | . . . . . 6 ⊢ (;11 + ;49) = ;60 |
| 27 | 11, 26 | eqtri 2765 | . . . . 5 ⊢ (;11 + (7 · 7)) = ;60 |
| 28 | 27 | oveq2i 7442 | . . . 4 ⊢ (7 gcd (;11 + (7 · 7))) = (7 gcd ;60) |
| 29 | 9, 28 | eqtri 2765 | . . 3 ⊢ (7 gcd ;11) = (7 gcd ;60) |
| 30 | 7re 12359 | . . . . . 6 ⊢ 7 ∈ ℝ | |
| 31 | 1 | nnnn0i 12534 | . . . . . . . 8 ⊢ 7 ∈ ℕ0 |
| 32 | 31 | dec0h 12755 | . . . . . . 7 ⊢ 7 = ;07 |
| 33 | 0nn0 12541 | . . . . . . . 8 ⊢ 0 ∈ ℕ0 | |
| 34 | 7lt9 12466 | . . . . . . . . 9 ⊢ 7 < 9 | |
| 35 | 9re 12365 | . . . . . . . . . . 11 ⊢ 9 ∈ ℝ | |
| 36 | 30, 35 | pm3.2i 470 | . . . . . . . . . 10 ⊢ (7 ∈ ℝ ∧ 9 ∈ ℝ) |
| 37 | ltle 11349 | . . . . . . . . . 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 11785 | . . . . . . . 8 ⊢ 0 < 1 | |
| 41 | 33, 5, 31, 5, 39, 40 | declth 12763 | . . . . . . 7 ⊢ ;07 < ;11 |
| 42 | 32, 41 | eqbrtri 5164 | . . . . . 6 ⊢ 7 < ;11 |
| 43 | ltne 11358 | . . . . . 6 ⊢ ((7 ∈ ℝ ∧ 7 < ;11) → ;11 ≠ 7) | |
| 44 | 30, 42, 43 | mp2an 692 | . . . . 5 ⊢ ;11 ≠ 7 |
| 45 | necom 2994 | . . . . 5 ⊢ (7 ≠ ;11 ↔ ;11 ≠ 7) | |
| 46 | 44, 45 | mpbir 231 | . . . 4 ⊢ 7 ≠ ;11 |
| 47 | 7prm 17148 | . . . . 5 ⊢ 7 ∈ ℙ | |
| 48 | 11prm 17152 | . . . . 5 ⊢ ;11 ∈ ℙ | |
| 49 | prmrp 16749 | . . . . 5 ⊢ ((7 ∈ ℙ ∧ ;11 ∈ ℙ) → ((7 gcd ;11) = 1 ↔ 7 ≠ ;11)) | |
| 50 | 47, 48, 49 | mp2an 692 | . . . 4 ⊢ ((7 gcd ;11) = 1 ↔ 7 ≠ ;11) |
| 51 | 46, 50 | mpbir 231 | . . 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 206 ∧ wa 395 = 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 ≤ cle 11296 4c4 12323 5c5 12324 6c6 12325 7c7 12326 9c9 12328 ;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: 60lcm7e420 42011 |
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