Theorem zrei 12535

Description: An integer is a real number. (Contributed by NM, 14-Jul-2005.)

Hypothesis

Ref	Expression
zrei.1	⊢ 𝐴 ∈ ℤ

Assertion

Ref	Expression
zrei	⊢ 𝐴 ∈ ℝ

Proof of Theorem zrei

Step	Hyp	Ref	Expression
1		zrei.1	. 2 ⊢ 𝐴 ∈ ℤ
2		zre 12533	. 2 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ

Syntax hints:   ∈ wcel 2109  ℝcr 11067  ℤcz 12529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-neg 11408  df-z 12530

This theorem is referenced by:  dfuzi  12625  eluzaddiOLD  12825  eluzsubiOLD  12827  dvdslelem  16279  divalglem1  16364  divalglem6  16368  divalglem9  16371  gcdaddmlem  16494  basellem9  26999  axlowdimlem16  28884  poimirlem17  37631  poimirlem19  37633  poimirlem20  37634  fdc  37739  jm2.27dlem2  42999