Theorem zrei 12619

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 12617	. 2 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ ℝ

Syntax hints:   ∈ wcel 2108  ℝcr 11154  ℤcz 12613

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-neg 11495  df-z 12614

This theorem is referenced by:  dfuzi  12709  eluzaddiOLD  12910  eluzsubiOLD  12912  dvdslelem  16346  divalglem1  16431  divalglem6  16435  divalglem9  16438  gcdaddmlem  16561  basellem9  27132  axlowdimlem16  28972  poimirlem17  37644  poimirlem19  37646  poimirlem20  37647  fdc  37752  jm2.27dlem2  43022