Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > iseven | Structured version Visualization version GIF version |
Description: The predicate "is an even number". An even number is an integer which is divisible by 2, i.e. the result of dividing the even integer by 2 is still an integer. (Contributed by AV, 14-Jun-2020.) |
Ref | Expression |
---|---|
iseven | ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7157 | . . 3 ⊢ (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2)) | |
2 | 1 | eleq1d 2897 | . 2 ⊢ (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ)) |
3 | df-even 43785 | . 2 ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ} | |
4 | 2, 3 | elrab2 3682 | 1 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 / cdiv 11291 2c2 11686 ℤcz 11975 Even ceven 43783 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-even 43785 |
This theorem is referenced by: evenz 43789 evendiv2z 43791 evenm1odd 43798 evenp1odd 43799 oddp1eveni 43800 oddm1eveni 43801 evennodd 43802 oddneven 43803 enege 43804 zeoALTV 43829 oddm1evenALTV 43834 oddp1evenALTV 43835 0evenALTV 43847 2evenALTV 43851 6even 43870 8even 43872 |
Copyright terms: Public domain | W3C validator |