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Mathbox for Alexander van der Vekens |
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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 47553 | . 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 2106 (class class class)co 7431 / cdiv 11918 2c2 12319 ℤcz 12611 Even ceven 47549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-even 47551 |
This theorem is referenced by: evenm1odd 47564 evenp1odd 47565 bits0eALTV 47605 opeoALTV 47609 omeoALTV 47611 epoo 47628 emoo 47629 epee 47630 emee 47631 evensumeven 47632 evenltle 47642 even3prm2 47644 mogoldbblem 47645 sbgoldbalt 47706 sgoldbeven3prm 47708 mogoldbb 47710 bgoldbachlt 47738 tgblthelfgott 47740 |
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