Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iseven Structured version   Visualization version   GIF version

Theorem iseven 47200
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.)
Assertion
Ref Expression
iseven (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))

Proof of Theorem iseven
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 oveq1 7431 . . 3 (𝑧 = 𝑍 → (𝑧 / 2) = (𝑍 / 2))
21eleq1d 2811 . 2 (𝑧 = 𝑍 → ((𝑧 / 2) ∈ ℤ ↔ (𝑍 / 2) ∈ ℤ))
3 df-even 47198 . 2 Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
42, 3elrab2 3684 1 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wcel 2099  (class class class)co 7424   / cdiv 11921  2c2 12319  cz 12610   Even ceven 47196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562  df-ov 7427  df-even 47198
This theorem is referenced by:  evenz  47202  evendiv2z  47204  evenm1odd  47211  evenp1odd  47212  oddp1eveni  47213  oddm1eveni  47214  evennodd  47215  oddneven  47216  enege  47217  zeoALTV  47242  oddm1evenALTV  47247  oddp1evenALTV  47248  0evenALTV  47260  2evenALTV  47264  6even  47283  8even  47285
  Copyright terms: Public domain W3C validator