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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 47105 | . 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) | |
2 | 1 | simplbi 496 | 1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 (class class class)co 7419 / cdiv 11903 2c2 12300 ℤcz 12591 Even ceven 47101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-even 47103 |
This theorem is referenced by: evenm1odd 47116 evenp1odd 47117 bits0eALTV 47157 opeoALTV 47161 omeoALTV 47163 epoo 47180 emoo 47181 epee 47182 emee 47183 evensumeven 47184 evenltle 47194 even3prm2 47196 mogoldbblem 47197 sbgoldbalt 47258 sgoldbeven3prm 47260 mogoldbb 47262 bgoldbachlt 47290 tgblthelfgott 47292 |
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