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	|- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ

Proof

Step	Hyp	Ref	Expression
1		sqrt2irr0	|- ( sqrt ` 2 ) e. ( RR \ QQ )
2		2logb9irr	|- ( 2 logb 9 ) e. ( RR \ QQ )
3		sqrt2cxp2logb9e3	|- ( ( sqrt ` 2 ) ^c ( 2 logb 9 ) ) = 3
4		3z	|- 3 e. ZZ
5		zq	|- ( 3 e. ZZ -> 3 e. QQ )
6	4 5	ax-mp	|- 3 e. QQ
7	3 6	eqeltri	|- ( ( sqrt ` 2 ) ^c ( 2 logb 9 ) ) e. QQ
8		oveq1	|- ( a = ( sqrt ` 2 ) -> ( a ^c b ) = ( ( sqrt ` 2 ) ^c b ) )
9	8	eleq1d	|- ( a = ( sqrt ` 2 ) -> ( ( a ^c b ) e. QQ <-> ( ( sqrt ` 2 ) ^c b ) e. QQ ) )
10		oveq2	|- ( b = ( 2 logb 9 ) -> ( ( sqrt ` 2 ) ^c b ) = ( ( sqrt ` 2 ) ^c ( 2 logb 9 ) ) )
11	10	eleq1d	|- ( b = ( 2 logb 9 ) -> ( ( ( sqrt ` 2 ) ^c b ) e. QQ <-> ( ( sqrt ` 2 ) ^c ( 2 logb 9 ) ) e. QQ ) )
12	9 11	rspc2ev	|- ( ( ( sqrt ` 2 ) e. ( RR \ QQ ) /\ ( 2 logb 9 ) e. ( RR \ QQ ) /\ ( ( sqrt ` 2 ) ^c ( 2 logb 9 ) ) e. QQ ) -> E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ )
13	1 2 7 12	mp3an	|- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ