![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > gcd2odd1 | Structured version Visualization version GIF version |
Description: The greatest common divisor of an odd number and 2 is 1, i.e., 2 and any odd number are coprime. Remark: The proof using dfodd7 47054 is longer (see proof in comment)! (Contributed by AV, 5-Jun-2023.) |
Ref | Expression |
---|---|
gcd2odd1 | ⊢ (𝑍 ∈ Odd → (𝑍 gcd 2) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddz 47018 | . . 3 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ) | |
2 | 2z 12634 | . . 3 ⊢ 2 ∈ ℤ | |
3 | gcdcom 16497 | . . 3 ⊢ ((𝑍 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑍 gcd 2) = (2 gcd 𝑍)) | |
4 | 1, 2, 3 | sylancl 584 | . 2 ⊢ (𝑍 ∈ Odd → (𝑍 gcd 2) = (2 gcd 𝑍)) |
5 | 2ndvdsodd 47042 | . . 3 ⊢ (𝑍 ∈ Odd → ¬ 2 ∥ 𝑍) | |
6 | 2prm 16672 | . . . 4 ⊢ 2 ∈ ℙ | |
7 | coprm 16691 | . . . 4 ⊢ ((2 ∈ ℙ ∧ 𝑍 ∈ ℤ) → (¬ 2 ∥ 𝑍 ↔ (2 gcd 𝑍) = 1)) | |
8 | 6, 1, 7 | sylancr 585 | . . 3 ⊢ (𝑍 ∈ Odd → (¬ 2 ∥ 𝑍 ↔ (2 gcd 𝑍) = 1)) |
9 | 5, 8 | mpbid 231 | . 2 ⊢ (𝑍 ∈ Odd → (2 gcd 𝑍) = 1) |
10 | 4, 9 | eqtrd 2768 | 1 ⊢ (𝑍 ∈ Odd → (𝑍 gcd 2) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 1c1 11149 2c2 12307 ℤcz 12598 ∥ cdvds 16240 gcd cgcd 16478 ℙcprime 16651 Odd codd 47012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-inf 9476 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-rp 13017 df-fz 13527 df-seq 14009 df-exp 14069 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-dvds 16241 df-gcd 16479 df-prm 16652 df-odd 47014 |
This theorem is referenced by: fpprel2 47128 |
Copyright terms: Public domain | W3C validator |