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 6657 | . . 3 ⊢ (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2)) | |
2 | 1 | eleq1d 2686 | . 2 ⊢ (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ)) |
3 | df-even 41539 | . 2 ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ} | |
4 | 2, 3 | elrab2 3366 | 1 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 / cdiv 10684 2c2 11070 ℤcz 11377 Even ceven 41537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-even 41539 |
This theorem is referenced by: evenz 41543 evendiv2z 41545 evenm1odd 41552 evenp1odd 41553 oddp1eveni 41554 oddm1eveni 41555 evennodd 41556 oddneven 41557 enege 41558 zeoALTV 41581 oddm1evenALTV 41586 oddp1evenALTV 41587 0evenALTV 41599 2evenALTV 41603 6even 41620 8even 41622 |
Copyright terms: Public domain | W3C validator |