![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nn0ob | GIF version |
Description: Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.) |
Ref | Expression |
---|---|
nn0ob | ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℕ0 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0o 11350 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) | |
2 | nn0cn 8781 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
3 | xp1d2m1eqxm1d2 8766 | . . . . . . 7 ⊢ (𝑁 ∈ ℂ → (((𝑁 + 1) / 2) − 1) = ((𝑁 − 1) / 2)) | |
4 | 3 | eqcomd 2100 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) / 2) = (((𝑁 + 1) / 2) − 1)) |
5 | 2, 4 | syl 14 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) / 2) = (((𝑁 + 1) / 2) − 1)) |
6 | peano2cnm 7845 | . . . . . . . 8 ⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) | |
7 | 2, 6 | syl 14 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 1) ∈ ℂ) |
8 | 7 | halfcld 8758 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) / 2) ∈ ℂ) |
9 | 1cnd 7601 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) | |
10 | peano2nn0 8811 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
11 | 10 | nn0cnd 8826 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℂ) |
12 | 11 | halfcld 8758 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) / 2) ∈ ℂ) |
13 | 8, 9, 12 | addlsub 7945 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((((𝑁 − 1) / 2) + 1) = ((𝑁 + 1) / 2) ↔ ((𝑁 − 1) / 2) = (((𝑁 + 1) / 2) − 1))) |
14 | 5, 13 | mpbird 166 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 − 1) / 2) + 1) = ((𝑁 + 1) / 2)) |
15 | 14 | adantr 271 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 − 1) / 2) ∈ ℕ0) → (((𝑁 − 1) / 2) + 1) = ((𝑁 + 1) / 2)) |
16 | peano2nn0 8811 | . . . 4 ⊢ (((𝑁 − 1) / 2) ∈ ℕ0 → (((𝑁 − 1) / 2) + 1) ∈ ℕ0) | |
17 | 16 | adantl 272 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 − 1) / 2) ∈ ℕ0) → (((𝑁 − 1) / 2) + 1) ∈ ℕ0) |
18 | 15, 17 | eqeltrrd 2172 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 − 1) / 2) ∈ ℕ0) → ((𝑁 + 1) / 2) ∈ ℕ0) |
19 | 1, 18 | impbida 564 | 1 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℕ0 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 ∈ wcel 1445 (class class class)co 5690 ℂcc 7445 1c1 7448 + caddc 7450 − cmin 7750 / cdiv 8236 2c2 8571 ℕ0cn0 8771 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-3 8580 df-4 8581 df-n0 8772 df-z 8849 df-uz 9119 |
This theorem is referenced by: nn0oddm1d2 11352 |
Copyright terms: Public domain | W3C validator |