Theorem 2sqreu

Description: There exists a unique decomposition of a prime of the form 4 k + 1 as a sum of squares of two nonnegative integers. See 2sqnn0 for the existence of such a decomposition. (Contributed by AV, 4-Jun-2023) (Revised by AV, 25-Jun-2023)

	Ref	Expression
Hypothesis	2sqreu.1	|- ( ph <-> ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) )
Assertion	2sqreu	|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( E! a e. NN0 E. b e. NN0 ph /\ E! b e. NN0 E. a e. NN0 ph ) )

Proof

Step	Hyp	Ref	Expression
1		2sqreu.1	|- ( ph <-> ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) )
2		2sqreulem1	|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> E! a e. NN0 E! b e. NN0 ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) )
3	1	bicomi	|- ( ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> ph )
4	3	reubii	|- ( E! b e. NN0 ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> E! b e. NN0 ph )
5	4	reubii	|- ( E! a e. NN0 E! b e. NN0 ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> E! a e. NN0 E! b e. NN0 ph )
6	1	2sqreulem4	|- A. a e. NN0 E* b e. NN0 ph
7		2reu1	|- ( A. a e. NN0 E* b e. NN0 ph -> ( E! a e. NN0 E! b e. NN0 ph <-> ( E! a e. NN0 E. b e. NN0 ph /\ E! b e. NN0 E. a e. NN0 ph ) ) )
8	6 7	mp1i	|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( E! a e. NN0 E! b e. NN0 ph <-> ( E! a e. NN0 E. b e. NN0 ph /\ E! b e. NN0 E. a e. NN0 ph ) ) )
9	5 8	syl5bb	|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( E! a e. NN0 E! b e. NN0 ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> ( E! a e. NN0 E. b e. NN0 ph /\ E! b e. NN0 E. a e. NN0 ph ) ) )
10	2 9	mpbid	|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( E! a e. NN0 E. b e. NN0 ph /\ E! b e. NN0 E. a e. NN0 ph ) )