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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 8905 | . . . . . 6 ⊢ 0 ≠ 1 | |
2 | pm13.181 2409 | . . . . . 6 ⊢ (((𝑁 mod 2) = 0 ∧ 0 ≠ 1) → (𝑁 mod 2) ≠ 1) | |
3 | 1, 2 | mpan2 422 | . . . . 5 ⊢ ((𝑁 mod 2) = 0 → (𝑁 mod 2) ≠ 1) |
4 | 3 | neneqd 2348 | . . . 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 11781 | . . . . 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 620 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 2) = 0) → ¬ ¬ 2 ∥ 𝑁) |
9 | 5, 8 | 2falsed 692 | . 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 8906 | . . . . . . 7 ⊢ 1 ≠ 0 | |
12 | pm13.181 2409 | . . . . . . 7 ⊢ (((𝑁 mod 2) = 1 ∧ 1 ≠ 0) → (𝑁 mod 2) ≠ 0) | |
13 | 11, 12 | mpan2 422 | . . . . . 6 ⊢ ((𝑁 mod 2) = 1 → (𝑁 mod 2) ≠ 0) |
14 | 13 | neneqd 2348 | . . . . 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 657 | . . . . 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 8999 | . . . . 5 ⊢ 2 ∈ ℕ | |
21 | zmodfz 10254 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑁 mod 2) ∈ (0...(2 − 1))) | |
22 | 20, 21 | mpan2 422 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...(2 − 1))) |
23 | 2m1e1 8956 | . . . . 5 ⊢ (2 − 1) = 1 | |
24 | 23 | oveq2i 5837 | . . . 4 ⊢ (0...(2 − 1)) = (0...1) |
25 | 22, 24 | eleqtrdi 2250 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...1)) |
26 | fz01or 10019 | . . 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 788 | 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 698 = wceq 1335 ∈ wcel 2128 ≠ wne 2327 class class class wbr 3967 (class class class)co 5826 0cc0 7734 1c1 7735 − cmin 8050 ℕcn 8838 2c2 8889 ℤcz 9172 ...cfz 9918 mod cmo 10230 ∥ cdvds 11694 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 ax-arch 7853 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-po 4258 df-iso 4259 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 df-inn 8839 df-2 8897 df-n0 9096 df-z 9173 df-uz 9445 df-q 9535 df-rp 9567 df-fz 9919 df-fl 10178 df-mod 10231 df-dvds 11695 |
This theorem is referenced by: (None) |
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