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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 8811 | . . . . . 6 ⊢ 0 ≠ 1 | |
2 | pm13.181 2391 | . . . . . 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 2330 | . . . 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 11611 | . . . . 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 8812 | . . . . . . 7 ⊢ 1 ≠ 0 | |
12 | pm13.181 2391 | . . . . . . 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 2330 | . . . . 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 8905 | . . . . 5 ⊢ 2 ∈ ℕ | |
21 | zmodfz 10150 | . . . . 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 8862 | . . . . 5 ⊢ (2 − 1) = 1 | |
24 | 23 | oveq2i 5793 | . . . 4 ⊢ (0...(2 − 1)) = (0...1) |
25 | 22, 24 | eleqtrdi 2233 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 mod 2) ∈ (0...1)) |
26 | fz01or 9922 | . . 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 1332 ∈ wcel 1481 ≠ wne 2309 class class class wbr 3937 (class class class)co 5782 0cc0 7644 1c1 7645 − cmin 7957 ℕcn 8744 2c2 8795 ℤcz 9078 ...cfz 9821 mod cmo 10126 ∥ cdvds 11529 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-fz 9822 df-fl 10074 df-mod 10127 df-dvds 11530 |
This theorem is referenced by: (None) |
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