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Mirrors > Home > ILE Home > Th. List > mod2eq1n2dvds | GIF version |
Description: An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.) |
Ref | Expression |
---|---|
mod2eq1n2dvds | ⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ne1 8550 | . . . . . 6 ⊢ 0 ≠ 1 | |
2 | pm13.181 2338 | . . . . . 6 ⊢ (((𝑁 mod 2) = 0 ∧ 0 ≠ 1) → (𝑁 mod 2) ≠ 1) | |
3 | 1, 2 | mpan2 417 | . . . . 5 ⊢ ((𝑁 mod 2) = 0 → (𝑁 mod 2) ≠ 1) |
4 | 3 | neneqd 2277 | . . . 4 ⊢ ((𝑁 mod 2) = 0 → ¬ (𝑁 mod 2) = 1) |
5 | 4 | adantl 272 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ¬ (𝑁 mod 2) = 1) |
6 | mod2eq0even 11217 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ↔ 2 ∥ 𝑁)) | |
7 | 6 | biimpa 291 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → 2 ∥ 𝑁) |
8 | 7 | notnotd 596 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ¬ ¬ 2 ∥ 𝑁) |
9 | 5, 8 | 2falsed 654 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁)) |
10 | simpr 109 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → (𝑁 mod 2) = 1) | |
11 | 1ne0 8551 | . . . . . . 7 ⊢ 1 ≠ 0 | |
12 | pm13.181 2338 | . . . . . . 7 ⊢ (((𝑁 mod 2) = 1 ∧ 1 ≠ 0) → (𝑁 mod 2) ≠ 0) | |
13 | 11, 12 | mpan2 417 | . . . . . 6 ⊢ ((𝑁 mod 2) = 1 → (𝑁 mod 2) ≠ 0) |
14 | 13 | neneqd 2277 | . . . . 5 ⊢ ((𝑁 mod 2) = 1 → ¬ (𝑁 mod 2) = 0) |
15 | 14 | adantl 272 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → ¬ (𝑁 mod 2) = 0) |
16 | 6 | notbid 628 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (¬ (𝑁 mod 2) = 0 ↔ ¬ 2 ∥ 𝑁)) |
17 | 16 | adantr 271 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → (¬ (𝑁 mod 2) = 0 ↔ ¬ 2 ∥ 𝑁)) |
18 | 15, 17 | mpbid 146 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → ¬ 2 ∥ 𝑁) |
19 | 10, 18 | 2thd 174 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁)) |
20 | 2nn 8638 | . . . . 5 ⊢ 2 ∈ ℕ | |
21 | zmodfz 9814 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑁 mod 2) ∈ (0...(2 − 1))) | |
22 | 20, 21 | mpan2 417 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...(2 − 1))) |
23 | 2m1e1 8601 | . . . . 5 ⊢ (2 − 1) = 1 | |
24 | 23 | oveq2i 5677 | . . . 4 ⊢ (0...(2 − 1)) = (0...1) |
25 | 22, 24 | syl6eleq 2181 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...1)) |
26 | fz01or 9586 | . . 3 ⊢ ((𝑁 mod 2) ∈ (0...1) ↔ ((𝑁 mod 2) = 0 ∨ (𝑁 mod 2) = 1)) | |
27 | 25, 26 | sylib 121 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ∨ (𝑁 mod 2) = 1)) |
28 | 9, 19, 27 | mpjaodan 748 | 1 ⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 665 = wceq 1290 ∈ wcel 1439 ≠ wne 2256 class class class wbr 3851 (class class class)co 5666 0cc0 7411 1c1 7412 − cmin 7714 ℕcn 8483 2c2 8534 ℤcz 8811 ...cfz 9485 mod cmo 9790 ∥ cdvds 11135 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 ax-arch 7525 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-po 4132 df-iso 4133 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 df-inn 8484 df-2 8542 df-n0 8735 df-z 8812 df-uz 9081 df-q 9166 df-rp 9196 df-fz 9486 df-fl 9738 df-mod 9791 df-dvds 11136 |
This theorem is referenced by: (None) |
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