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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 12006 | . . . . 5 ⊢ 1 ∈ ℤ | |
2 | gcddvds 15846 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑀 gcd 1) ∥ 𝑀 ∧ (𝑀 gcd 1) ∥ 1)) | |
3 | 1, 2 | mpan2 689 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑀 gcd 1) ∥ 𝑀 ∧ (𝑀 gcd 1) ∥ 1)) |
4 | 3 | simprd 498 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) ∥ 1) |
5 | ax-1ne0 10600 | . . . . . . . 8 ⊢ 1 ≠ 0 | |
6 | simpr 487 | . . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 1 = 0) → 1 = 0) | |
7 | 6 | necon3ai 3041 | . . . . . . . 8 ⊢ (1 ≠ 0 → ¬ (𝑀 = 0 ∧ 1 = 0)) |
8 | 5, 7 | ax-mp 5 | . . . . . . 7 ⊢ ¬ (𝑀 = 0 ∧ 1 = 0) |
9 | gcdn0cl 15845 | . . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 1 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 1 = 0)) → (𝑀 gcd 1) ∈ ℕ) | |
10 | 8, 9 | mpan2 689 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑀 gcd 1) ∈ ℕ) |
11 | 1, 10 | mpan2 689 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) ∈ ℕ) |
12 | 11 | nnzd 12080 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) ∈ ℤ) |
13 | 1nn 11643 | . . . 4 ⊢ 1 ∈ ℕ | |
14 | dvdsle 15654 | . . . 4 ⊢ (((𝑀 gcd 1) ∈ ℤ ∧ 1 ∈ ℕ) → ((𝑀 gcd 1) ∥ 1 → (𝑀 gcd 1) ≤ 1)) | |
15 | 12, 13, 14 | sylancl 588 | . . 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 11661 | . . 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 234 | 1 ⊢ (𝑀 ∈ ℤ → (𝑀 gcd 1) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 (class class class)co 7150 0cc0 10531 1c1 10532 ≤ cle 10670 ℕcn 11632 ℤcz 11975 ∥ cdvds 15601 gcd cgcd 15837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 df-gcd 15838 |
This theorem is referenced by: 1gcd 15875 lcm1 15948 dfphi2 16105 pockthlem 16235 fvprmselgcd1 16375 odinv 18682 pgpfac1lem2 19191 lgs1 25911 lgsquad2lem2 25955 2sqlem11 25999 qqh1 31221 nn0expgcd 39177 |
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