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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 8945 | . . . . . 6 ⊢ 0 ≠ 1 | |
2 | pm13.181 2422 | . . . . . 6 ⊢ (((𝑁 mod 2) = 0 ∧ 0 ≠ 1) → (𝑁 mod 2) ≠ 1) | |
3 | 1, 2 | mpan2 423 | . . . . 5 ⊢ ((𝑁 mod 2) = 0 → (𝑁 mod 2) ≠ 1) |
4 | 3 | neneqd 2361 | . . . 4 ⊢ ((𝑁 mod 2) = 0 → ¬ (𝑁 mod 2) = 1) |
5 | 4 | adantl 275 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ¬ (𝑁 mod 2) = 1) |
6 | mod2eq0even 11837 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ↔ 2 ∥ 𝑁)) | |
7 | 6 | biimpa 294 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → 2 ∥ 𝑁) |
8 | 7 | notnotd 625 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ¬ ¬ 2 ∥ 𝑁) |
9 | 5, 8 | 2falsed 697 | . 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 8946 | . . . . . . 7 ⊢ 1 ≠ 0 | |
12 | pm13.181 2422 | . . . . . . 7 ⊢ (((𝑁 mod 2) = 1 ∧ 1 ≠ 0) → (𝑁 mod 2) ≠ 0) | |
13 | 11, 12 | mpan2 423 | . . . . . 6 ⊢ ((𝑁 mod 2) = 1 → (𝑁 mod 2) ≠ 0) |
14 | 13 | neneqd 2361 | . . . . 5 ⊢ ((𝑁 mod 2) = 1 → ¬ (𝑁 mod 2) = 0) |
15 | 14 | adantl 275 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → ¬ (𝑁 mod 2) = 0) |
16 | 6 | notbid 662 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (¬ (𝑁 mod 2) = 0 ↔ ¬ 2 ∥ 𝑁)) |
17 | 16 | adantr 274 | . . . 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 9039 | . . . . 5 ⊢ 2 ∈ ℕ | |
21 | zmodfz 10302 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑁 mod 2) ∈ (0...(2 − 1))) | |
22 | 20, 21 | mpan2 423 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...(2 − 1))) |
23 | 2m1e1 8996 | . . . . 5 ⊢ (2 − 1) = 1 | |
24 | 23 | oveq2i 5864 | . . . 4 ⊢ (0...(2 − 1)) = (0...1) |
25 | 22, 24 | eleqtrdi 2263 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...1)) |
26 | fz01or 10067 | . . 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 793 | 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 703 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 class class class wbr 3989 (class class class)co 5853 0cc0 7774 1c1 7775 − cmin 8090 ℕcn 8878 2c2 8929 ℤcz 9212 ...cfz 9965 mod cmo 10278 ∥ cdvds 11749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fl 10226 df-mod 10279 df-dvds 11750 |
This theorem is referenced by: (None) |
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