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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evendiv2z | Structured version Visualization version GIF version |
Description: The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.) |
Ref | Expression |
---|---|
evendiv2z | ⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseven 45791 | . 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) | |
2 | 1 | simprbi 497 | 1 ⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 (class class class)co 7356 / cdiv 11811 2c2 12207 ℤcz 12498 Even ceven 45787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-iota 6448 df-fv 6504 df-ov 7359 df-even 45789 |
This theorem is referenced by: zefldiv2ALTV 45824 nn0e 45860 nneven 45861 |
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