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Metamath Proof Explorer

Theorem 2irrexpqALT

Description: Alternate proof of 2irrexpq : There exist irrational numbers a and b such that ( a ^ b ) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of Bauer, p. 483. In contrast to 2irrexpq , this is a constructive proof because it is based on two explicitly named irrational numbers ( sqrt2 ) and ( 2 logb 9 ) , see sqrt2irr0 , 2logb9irr and sqrt2cxp2logb9e3 . Therefore, this proof is also acceptable/usable in intuitionistic logic. (Contributed by AV, 23-Dec-2022) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion 2irrexpqALT ∃ 𝑎 ∈ ( ℝ ∖ ℚ ) ∃ 𝑏 ∈ ( ℝ ∖ ℚ ) ( 𝑎 ↑𝑐 𝑏 ) ∈ ℚ

Proof

Step Hyp Ref Expression
1 sqrt2irr0 ⊢ ( √ ‘ 2 ) ∈ ( ℝ ∖ ℚ )
2 2logb9irr ⊢ ( 2 logb 9 ) ∈ ( ℝ ∖ ℚ )
3 sqrt2cxp2logb9e3 ⊢ ( ( √ ‘ 2 ) ↑𝑐 ( 2 logb 9 ) ) = 3
4 3z ⊢ 3 ∈ ℤ
5 zq ⊢ ( 3 ∈ ℤ → 3 ∈ ℚ )
6 4 5 ax-mp ⊢ 3 ∈ ℚ
7 3 6 eqeltri ⊢ ( ( √ ‘ 2 ) ↑𝑐 ( 2 logb 9 ) ) ∈ ℚ
8 oveq1 ⊢ ( 𝑎 = ( √ ‘ 2 ) → ( 𝑎 ↑𝑐 𝑏 ) = ( ( √ ‘ 2 ) ↑𝑐 𝑏 ) )
9 8 eleq1d ⊢ ( 𝑎 = ( √ ‘ 2 ) → ( ( 𝑎 ↑𝑐 𝑏 ) ∈ ℚ ↔ ( ( √ ‘ 2 ) ↑𝑐 𝑏 ) ∈ ℚ ) )
10 oveq2 ⊢ ( 𝑏 = ( 2 logb 9 ) → ( ( √ ‘ 2 ) ↑𝑐 𝑏 ) = ( ( √ ‘ 2 ) ↑𝑐 ( 2 logb 9 ) ) )
11 10 eleq1d ⊢ ( 𝑏 = ( 2 logb 9 ) → ( ( ( √ ‘ 2 ) ↑𝑐 𝑏 ) ∈ ℚ ↔ ( ( √ ‘ 2 ) ↑𝑐 ( 2 logb 9 ) ) ∈ ℚ ) )
12 9 11 rspc2ev ⊢ ( ( ( √ ‘ 2 ) ∈ ( ℝ ∖ ℚ ) ∧ ( 2 logb 9 ) ∈ ( ℝ ∖ ℚ ) ∧ ( ( √ ‘ 2 ) ↑𝑐 ( 2 logb 9 ) ) ∈ ℚ ) → ∃ 𝑎 ∈ ( ℝ ∖ ℚ ) ∃ 𝑏 ∈ ( ℝ ∖ ℚ ) ( 𝑎 ↑𝑐 𝑏 ) ∈ ℚ )
13 1 2 7 12 mp3an ⊢ ∃ 𝑎 ∈ ( ℝ ∖ ℚ ) ∃ 𝑏 ∈ ( ℝ ∖ ℚ ) ( 𝑎 ↑𝑐 𝑏 ) ∈ ℚ