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Metamath Proof Explorer

Theorem nzadd

Description: The sum of a real number not being an integer and an integer is not an integer. (Contributed by AV, 19-Jul-2021)

Ref Expression
Assertion nzadd ( ( 𝐴 ∈ ( ℝ ∖ ℤ ) ∧ 𝐵 ∈ ℤ ) → ( 𝐴 + 𝐵 ) ∈ ( ℝ ∖ ℤ ) )

Proof

Step Hyp Ref Expression
1 eldif ⊢ ( 𝐴 ∈ ( ℝ ∖ ℤ ) ↔ ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) )
2 zre ⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℝ )
3 readdcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )
4 2 3 sylan2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )
5 4 adantlr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) ∧ 𝐵 ∈ ℤ ) → ( 𝐴 + 𝐵 ) ∈ ℝ )
6 zsubcl ⊢ ( ( ( 𝐴 + 𝐵 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 𝐵 ) − 𝐵 ) ∈ ℤ )
7 6 expcom ⊢ ( 𝐵 ∈ ℤ → ( ( 𝐴 + 𝐵 ) ∈ ℤ → ( ( 𝐴 + 𝐵 ) − 𝐵 ) ∈ ℤ ) )
8 7 adantl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 𝐵 ) ∈ ℤ → ( ( 𝐴 + 𝐵 ) − 𝐵 ) ∈ ℤ ) )
9 recn ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
10 zcn ⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℂ )
11 pncan ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 𝐵 ) = 𝐴 )
12 9 10 11 syl2an ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 𝐵 ) − 𝐵 ) = 𝐴 )
13 12 eleq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( ( 𝐴 + 𝐵 ) − 𝐵 ) ∈ ℤ ↔ 𝐴 ∈ ℤ ) )
14 8 13 sylibd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 𝐵 ) ∈ ℤ → 𝐴 ∈ ℤ ) )
15 14 con3d ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ) → ( ¬ 𝐴 ∈ ℤ → ¬ ( 𝐴 + 𝐵 ) ∈ ℤ ) )
16 15 ex ⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℤ → ( ¬ 𝐴 ∈ ℤ → ¬ ( 𝐴 + 𝐵 ) ∈ ℤ ) ) )
17 16 com23 ⊢ ( 𝐴 ∈ ℝ → ( ¬ 𝐴 ∈ ℤ → ( 𝐵 ∈ ℤ → ¬ ( 𝐴 + 𝐵 ) ∈ ℤ ) ) )
18 17 imp31 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) ∧ 𝐵 ∈ ℤ ) → ¬ ( 𝐴 + 𝐵 ) ∈ ℤ )
19 5 18 jca ⊢ ( ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℤ ) ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 𝐵 ) ∈ ℝ ∧ ¬ ( 𝐴 + 𝐵 ) ∈ ℤ ) )
20 1 19 sylanb ⊢ ( ( 𝐴 ∈ ( ℝ ∖ ℤ ) ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 𝐵 ) ∈ ℝ ∧ ¬ ( 𝐴 + 𝐵 ) ∈ ℤ ) )
21 eldif ⊢ ( ( 𝐴 + 𝐵 ) ∈ ( ℝ ∖ ℤ ) ↔ ( ( 𝐴 + 𝐵 ) ∈ ℝ ∧ ¬ ( 𝐴 + 𝐵 ) ∈ ℤ ) )
22 20 21 sylibr ⊢ ( ( 𝐴 ∈ ( ℝ ∖ ℤ ) ∧ 𝐵 ∈ ℤ ) → ( 𝐴 + 𝐵 ) ∈ ( ℝ ∖ ℤ ) )