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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 7149 | . . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1)) | |
2 | 1 | oveq1d 7157 | . . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2)) |
3 | 2 | eleq1d 2897 | . 2 ⊢ (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ)) |
4 | df-odd 43877 | . 2 ⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ} | |
5 | 3, 4 | elrab2 3674 | 1 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7142 1c1 10524 + caddc 10526 / cdiv 11283 2c2 11679 ℤcz 11968 Odd codd 43875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-iota 6300 df-fv 6349 df-ov 7145 df-odd 43877 |
This theorem is referenced by: oddz 43881 oddp1div2z 43883 isodd2 43885 evenm1odd 43889 evennodd 43893 oddneven 43894 onego 43896 zeoALTV 43920 oddp1evenALTV 43926 1oddALTV 43940 |
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