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| Mirrors > Home > MPE Home > Th. List > Mathboxes > isodd | Structured version Visualization version GIF version | ||
| Description: The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020.) |
| Ref | Expression |
|---|---|
| isodd | ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq1 7438 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1)) | |
| 2 | 1 | oveq1d 7446 | . . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2)) |
| 3 | 2 | eleq1d 2826 | . 2 ⊢ (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ)) |
| 4 | df-odd 47614 | . 2 ⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ} | |
| 5 | 3, 4 | elrab2 3695 | 1 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 1c1 11156 + caddc 11158 / cdiv 11920 2c2 12321 ℤcz 12613 Odd codd 47612 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-odd 47614 |
| This theorem is referenced by: oddz 47618 oddp1div2z 47620 isodd2 47622 evenm1odd 47626 evennodd 47630 oddneven 47631 onego 47633 zeoALTV 47657 oddp1evenALTV 47663 1oddALTV 47677 |
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