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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 47621 | . 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ)) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 (class class class)co 7432 1c1 11157 + caddc 11159 / cdiv 11921 2c2 12322 ℤcz 12615 Odd codd 47617 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-odd 47619 | 
| This theorem is referenced by: oddm1div2z 47626 oddp1eveni 47633 oddm1eveni 47634 m1expoddALTV 47640 2dvdsoddp1 47648 2dvdsoddm1 47649 zofldiv2ALTV 47654 oddflALTV 47655 gcd2odd1 47660 oexpnegALTV 47669 oexpnegnz 47670 bits0oALTV 47673 opoeALTV 47675 opeoALTV 47676 omoeALTV 47677 omeoALTV 47678 epoo 47695 emoo 47696 stgoldbwt 47768 sbgoldbwt 47769 sbgoldbst 47770 sbgoldbm 47776 bgoldbtbndlem1 47797 bgoldbtbndlem2 47798 bgoldbtbndlem3 47799 bgoldbtbndlem4 47800 bgoldbtbnd 47801 tgoldbach 47809 | 
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