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| Mirrors > Home > MPE Home > Th. List > Mathboxes > evenz | Structured version Visualization version GIF version | ||
| Description: An even number is an integer. (Contributed by AV, 14-Jun-2020.) | 
| Ref | Expression | 
|---|---|
| evenz | ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iseven 47615 | . 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7431 / cdiv 11920 2c2 12321 ℤcz 12613 Even ceven 47611 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-even 47613 | 
| This theorem is referenced by: evenm1odd 47626 evenp1odd 47627 bits0eALTV 47667 opeoALTV 47671 omeoALTV 47673 epoo 47690 emoo 47691 epee 47692 emee 47693 evensumeven 47694 evenltle 47704 even3prm2 47706 mogoldbblem 47707 sbgoldbalt 47768 sgoldbeven3prm 47770 mogoldbb 47772 bgoldbachlt 47800 tgblthelfgott 47802 | 
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