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Mirrors > Home > MPE Home > Th. List > Mathboxes > oddz | Structured version Visualization version GIF version |
Description: An odd number is an integer. (Contributed by AV, 14-Jun-2020.) |
Ref | Expression |
---|---|
oddz | ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isodd 42549 | . 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) | |
2 | 1 | simplbi 493 | 1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ 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 42545 |
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 42547 |
This theorem is referenced by: oddm1div2z 42554 oddp1eveni 42561 oddm1eveni 42562 m1expoddALTV 42568 2dvdsoddp1 42575 2dvdsoddm1 42576 zofldiv2ALTV 42581 oddflALTV 42582 oexpnegALTV 42595 oexpnegnz 42596 bits0oALTV 42599 opoeALTV 42601 opeoALTV 42602 omoeALTV 42603 omeoALTV 42604 epoo 42619 emoo 42620 stgoldbwt 42671 sbgoldbwt 42672 sbgoldbst 42673 sbgoldbm 42679 bgoldbtbndlem1 42700 bgoldbtbndlem2 42701 bgoldbtbndlem3 42702 bgoldbtbndlem4 42703 bgoldbtbnd 42704 tgoldbach 42712 |
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