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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 7437 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1)) | |
2 | 1 | oveq1d 7445 | . . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2)) |
3 | 2 | eleq1d 2823 | . 2 ⊢ (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ)) |
4 | df-odd 47551 | . 2 ⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ} | |
5 | 3, 4 | elrab2 3697 | 1 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 (class class class)co 7430 1c1 11153 + caddc 11155 / cdiv 11917 2c2 12318 ℤcz 12610 Odd codd 47549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 df-odd 47551 |
This theorem is referenced by: oddz 47555 oddp1div2z 47557 isodd2 47559 evenm1odd 47563 evennodd 47567 oddneven 47568 onego 47570 zeoALTV 47594 oddp1evenALTV 47600 1oddALTV 47614 |
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