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 ) )