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Mirrors > Home > MPE Home > Th. List > gcd1 | Structured version Visualization version GIF version |
Description: The gcd of a number with 1 is 1. Theorem 1.4(d)1 in [ApostolNT] p. 16. (Contributed by Mario Carneiro, 19-Feb-2014.) |
Ref | Expression |
---|---|
gcd1 | ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 12000 | . . . . 5 ⊢ 1 ∈ ℤ | |
2 | gcddvds 15842 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑀 gcd 1) ∥ 𝑀 ∧ (𝑀 gcd 1) ∥ 1)) | |
3 | 1, 2 | mpan2 690 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑀 gcd 1) ∥ 𝑀 ∧ (𝑀 gcd 1) ∥ 1)) |
4 | 3 | simprd 499 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) ∥ 1) |
5 | ax-1ne0 10595 | . . . . . . . 8 ⊢ 1 ≠ 0 | |
6 | simpr 488 | . . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 1 = 0) → 1 = 0) | |
7 | 6 | necon3ai 3012 | . . . . . . . 8 ⊢ (1 ≠ 0 → ¬ (𝑀 = 0 ∧ 1 = 0)) |
8 | 5, 7 | ax-mp 5 | . . . . . . 7 ⊢ ¬ (𝑀 = 0 ∧ 1 = 0) |
9 | gcdn0cl 15841 | . . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 1 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 1 = 0)) → (𝑀 gcd 1) ∈ ℕ) | |
10 | 8, 9 | mpan2 690 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑀 gcd 1) ∈ ℕ) |
11 | 1, 10 | mpan2 690 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) ∈ ℕ) |
12 | 11 | nnzd 12074 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) ∈ ℤ) |
13 | 1nn 11636 | . . . 4 ⊢ 1 ∈ ℕ | |
14 | dvdsle 15652 | . . . 4 ⊢ (((𝑀 gcd 1) ∈ ℤ ∧ 1 ∈ ℕ) → ((𝑀 gcd 1) ∥ 1 → (𝑀 gcd 1) ≤ 1)) | |
15 | 12, 13, 14 | sylancl 589 | . . 3 ⊢ (𝑀 ∈ ℤ → ((𝑀 gcd 1) ∥ 1 → (𝑀 gcd 1) ≤ 1)) |
16 | 4, 15 | mpd 15 | . 2 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) ≤ 1) |
17 | nnle1eq1 11655 | . . 3 ⊢ ((𝑀 gcd 1) ∈ ℕ → ((𝑀 gcd 1) ≤ 1 ↔ (𝑀 gcd 1) = 1)) | |
18 | 11, 17 | syl 17 | . 2 ⊢ (𝑀 ∈ ℤ → ((𝑀 gcd 1) ≤ 1 ↔ (𝑀 gcd 1) = 1)) |
19 | 16, 18 | mpbid 235 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 0cc0 10526 1c1 10527 ≤ cle 10665 ℕcn 11625 ℤcz 11969 ∥ cdvds 15599 gcd cgcd 15833 |
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-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 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-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 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 |
This theorem is referenced by: 1gcd 15871 lcm1 15944 dfphi2 16101 pockthlem 16231 fvprmselgcd1 16371 odinv 18680 pgpfac1lem2 19190 lgs1 25925 lgsquad2lem2 25969 2sqlem11 26013 qqh1 31336 lcmineqlem19 39335 nn0expgcd 39492 |
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