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Metamath Proof Explorer

Theorem irradd

Description: The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008)

Ref Expression
Assertion irradd ( ( 𝐴 ∈ ( ℝ ∖ ℚ ) ∧ 𝐵 ∈ ℚ ) → ( 𝐴 + 𝐵 ) ∈ ( ℝ ∖ ℚ ) )

Proof

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