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Theorem evenz 42071
Description: An even number is an integer. (Contributed by AV, 14-Jun-2020.)
Assertion
Ref Expression
evenz (𝑍 ∈ Even → 𝑍 ∈ ℤ)

Proof of Theorem evenz
StepHypRef Expression
1 iseven 42069 . 2 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
21simplbi 485 1 (𝑍 ∈ Even → 𝑍 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  (class class class)co 6793   / cdiv 10886  2c2 11272  cz 11579   Even ceven 42065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-even 42067
This theorem is referenced by:  evenm1odd  42080  evenp1odd  42081  bits0eALTV  42119  opeoALTV  42123  omeoALTV  42125  epoo  42140  emoo  42141  epee  42142  emee  42143  evensumeven  42144  evenltle  42154  even3prm2  42156  mogoldbblem  42157  sbgoldbalt  42197  sgoldbeven3prm  42199  mogoldbb  42201  bgoldbachlt  42229  tgblthelfgott  42231  bgoldbachltOLD  42235  tgblthelfgottOLD  42237
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