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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 8999 | . . . . . 6 ⊢ 0 ≠ 1 | |
2 | pm13.181 2439 | . . . . . 6 ⊢ (((𝑁 mod 2) = 0 ∧ 0 ≠ 1) → (𝑁 mod 2) ≠ 1) | |
3 | 1, 2 | mpan2 425 | . . . . 5 ⊢ ((𝑁 mod 2) = 0 → (𝑁 mod 2) ≠ 1) |
4 | 3 | neneqd 2378 | . . . 4 ⊢ ((𝑁 mod 2) = 0 → ¬ (𝑁 mod 2) = 1) |
5 | 4 | adantl 277 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ¬ (𝑁 mod 2) = 1) |
6 | mod2eq0even 11896 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ↔ 2 ∥ 𝑁)) | |
7 | 6 | biimpa 296 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → 2 ∥ 𝑁) |
8 | 7 | notnotd 631 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ¬ ¬ 2 ∥ 𝑁) |
9 | 5, 8 | 2falsed 703 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁)) |
10 | simpr 110 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → (𝑁 mod 2) = 1) | |
11 | 1ne0 9000 | . . . . . . 7 ⊢ 1 ≠ 0 | |
12 | pm13.181 2439 | . . . . . . 7 ⊢ (((𝑁 mod 2) = 1 ∧ 1 ≠ 0) → (𝑁 mod 2) ≠ 0) | |
13 | 11, 12 | mpan2 425 | . . . . . 6 ⊢ ((𝑁 mod 2) = 1 → (𝑁 mod 2) ≠ 0) |
14 | 13 | neneqd 2378 | . . . . 5 ⊢ ((𝑁 mod 2) = 1 → ¬ (𝑁 mod 2) = 0) |
15 | 14 | adantl 277 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → ¬ (𝑁 mod 2) = 0) |
16 | 6 | notbid 668 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (¬ (𝑁 mod 2) = 0 ↔ ¬ 2 ∥ 𝑁)) |
17 | 16 | adantr 276 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → (¬ (𝑁 mod 2) = 0 ↔ ¬ 2 ∥ 𝑁)) |
18 | 15, 17 | mpbid 147 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → ¬ 2 ∥ 𝑁) |
19 | 10, 18 | 2thd 175 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 1) → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁)) |
20 | 2nn 9093 | . . . . 5 ⊢ 2 ∈ ℕ | |
21 | zmodfz 10359 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑁 mod 2) ∈ (0...(2 − 1))) | |
22 | 20, 21 | mpan2 425 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...(2 − 1))) |
23 | 2m1e1 9050 | . . . . 5 ⊢ (2 − 1) = 1 | |
24 | 23 | oveq2i 5899 | . . . 4 ⊢ (0...(2 − 1)) = (0...1) |
25 | 22, 24 | eleqtrdi 2280 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...1)) |
26 | fz01or 10124 | . . 3 ⊢ ((𝑁 mod 2) ∈ (0...1) ↔ ((𝑁 mod 2) = 0 ∨ (𝑁 mod 2) = 1)) | |
27 | 25, 26 | sylib 122 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 0 ∨ (𝑁 mod 2) = 1)) |
28 | 9, 19, 27 | mpjaodan 799 | 1 ⊢ (𝑁 ∈ ℤ → ((𝑁 mod 2) = 1 ↔ ¬ 2 ∥ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 = wceq 1363 ∈ wcel 2158 ≠ wne 2357 class class class wbr 4015 (class class class)co 5888 0cc0 7824 1c1 7825 − cmin 8141 ℕcn 8932 2c2 8983 ℤcz 9266 ...cfz 10021 mod cmo 10335 ∥ cdvds 11807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-fz 10022 df-fl 10283 df-mod 10336 df-dvds 11808 |
This theorem is referenced by: (None) |
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