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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > oddp1div2z | Structured version Visualization version GIF version |
Description: The result of dividing an odd number increased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.) |
Ref | Expression |
---|---|
oddp1div2z | ⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isodd 42560 | . 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) | |
2 | 1 | simprbi 492 | 1 ⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 (class class class)co 6922 1c1 10273 + caddc 10275 / cdiv 11032 2c2 11430 ℤcz 11728 Odd codd 42556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-ov 6925 df-odd 42558 |
This theorem is referenced by: oddm1div2z 42565 oddp1eveni 42572 |
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