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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 | df-odd 40078 | . . 3 ⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ} | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (𝑍 ∈ Odd ↔ 𝑍 ∈ {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}) |
3 | oveq1 6556 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1)) | |
4 | 3 | oveq1d 6564 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2)) |
5 | 4 | eleq1d 2672 | . . 3 ⊢ (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ)) |
6 | 5 | elrab 3331 | . 2 ⊢ (𝑍 ∈ {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ} ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) |
7 | 2, 6 | bitri 263 | 1 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 (class class class)co 6549 1c1 9816 + caddc 9818 / cdiv 10563 2c2 10947 ℤcz 11254 Odd codd 40076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-odd 40078 |
This theorem is referenced by: oddz 40082 oddp1div2z 40084 isodd2 40086 evenm1odd 40090 evennodd 40094 oddneven 40095 onego 40097 zeoALTV 40119 oddp1evenALTV 40125 1oddALTV 40139 |
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