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Mirrors > Home > ILE Home > Th. List > zneo | GIF version |
Description: No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.) |
Ref | Expression |
---|---|
zneo | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ≠ ((2 · 𝐵) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfnz 8576 | . . 3 ⊢ ¬ (1 / 2) ∈ ℤ | |
2 | 2cn 8229 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
3 | zcn 8489 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
4 | 3 | adantr 270 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℂ) |
5 | mulcl 7214 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2 · 𝐴) ∈ ℂ) | |
6 | 2, 4, 5 | sylancr 405 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ∈ ℂ) |
7 | zcn 8489 | . . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
8 | 7 | adantl 271 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℂ) |
9 | mulcl 7214 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐵) ∈ ℂ) | |
10 | 2, 8, 9 | sylancr 405 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐵) ∈ ℂ) |
11 | 1cnd 7249 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 1 ∈ ℂ) | |
12 | 6, 10, 11 | subaddd 7556 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((2 · 𝐴) − (2 · 𝐵)) = 1 ↔ ((2 · 𝐵) + 1) = (2 · 𝐴))) |
13 | 2 | a1i 9 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 2 ∈ ℂ) |
14 | 13, 4, 8 | subdid 7637 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · (𝐴 − 𝐵)) = ((2 · 𝐴) − (2 · 𝐵))) |
15 | 14 | oveq1d 5578 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 · (𝐴 − 𝐵)) / 2) = (((2 · 𝐴) − (2 · 𝐵)) / 2)) |
16 | zsubcl 8525 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) ∈ ℤ) | |
17 | zcn 8489 | . . . . . . . . . 10 ⊢ ((𝐴 − 𝐵) ∈ ℤ → (𝐴 − 𝐵) ∈ ℂ) | |
18 | 16, 17 | syl 14 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) ∈ ℂ) |
19 | 2ap0 8251 | . . . . . . . . . 10 ⊢ 2 # 0 | |
20 | 19 | a1i 9 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 2 # 0) |
21 | 18, 13, 20 | divcanap3d 8001 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 · (𝐴 − 𝐵)) / 2) = (𝐴 − 𝐵)) |
22 | 15, 21 | eqtr3d 2117 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((2 · 𝐴) − (2 · 𝐵)) / 2) = (𝐴 − 𝐵)) |
23 | 22, 16 | eqeltrd 2159 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((2 · 𝐴) − (2 · 𝐵)) / 2) ∈ ℤ) |
24 | oveq1 5570 | . . . . . . 7 ⊢ (((2 · 𝐴) − (2 · 𝐵)) = 1 → (((2 · 𝐴) − (2 · 𝐵)) / 2) = (1 / 2)) | |
25 | 24 | eleq1d 2151 | . . . . . 6 ⊢ (((2 · 𝐴) − (2 · 𝐵)) = 1 → ((((2 · 𝐴) − (2 · 𝐵)) / 2) ∈ ℤ ↔ (1 / 2) ∈ ℤ)) |
26 | 23, 25 | syl5ibcom 153 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((2 · 𝐴) − (2 · 𝐵)) = 1 → (1 / 2) ∈ ℤ)) |
27 | 12, 26 | sylbird 168 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((2 · 𝐵) + 1) = (2 · 𝐴) → (1 / 2) ∈ ℤ)) |
28 | 27 | necon3bd 2292 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (¬ (1 / 2) ∈ ℤ → ((2 · 𝐵) + 1) ≠ (2 · 𝐴))) |
29 | 1, 28 | mpi 15 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 · 𝐵) + 1) ≠ (2 · 𝐴)) |
30 | 29 | necomd 2335 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ≠ ((2 · 𝐵) + 1)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ≠ wne 2249 class class class wbr 3805 (class class class)co 5563 ℂcc 7093 0cc0 7095 1c1 7096 + caddc 7098 · cmul 7100 − cmin 7398 # cap 7800 / cdiv 7879 2c2 8208 ℤcz 8484 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7181 ax-resscn 7182 ax-1cn 7183 ax-1re 7184 ax-icn 7185 ax-addcl 7186 ax-addrcl 7187 ax-mulcl 7188 ax-mulrcl 7189 ax-addcom 7190 ax-mulcom 7191 ax-addass 7192 ax-mulass 7193 ax-distr 7194 ax-i2m1 7195 ax-0lt1 7196 ax-1rid 7197 ax-0id 7198 ax-rnegex 7199 ax-precex 7200 ax-cnre 7201 ax-pre-ltirr 7202 ax-pre-ltwlin 7203 ax-pre-lttrn 7204 ax-pre-apti 7205 ax-pre-ltadd 7206 ax-pre-mulgt0 7207 ax-pre-mulext 7208 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2612 df-sbc 2825 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-br 3806 df-opab 3860 df-id 4076 df-po 4079 df-iso 4080 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-iota 4917 df-fun 4954 df-fv 4960 df-riota 5519 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-pnf 7269 df-mnf 7270 df-xr 7271 df-ltxr 7272 df-le 7273 df-sub 7400 df-neg 7401 df-reap 7794 df-ap 7801 df-div 7880 df-inn 8159 df-2 8217 df-n0 8408 df-z 8485 |
This theorem is referenced by: nneo 8583 zeo2 8586 |
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