Theorem 2sqreunnltb

Description: There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4 k + 1 . (Contributed by AV, 11-Jun-2023) The prime needs not be odd, as observed by WL. (Revised by AV, 25-Jun-2023)

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

Proof

Step	Hyp	Ref	Expression
1		2sqreult.1	|- ( ph <-> ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) )
2		2sqreunnltblem	|- ( P e. Prime -> ( ( P mod 4 ) = 1 <-> E! a e. NN E! b e. NN ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) )
3	1	bicomi	|- ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> ph )
4	3	reubii	|- ( E! b e. NN ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> E! b e. NN ph )
5	4	reubii	|- ( E! a e. NN E! b e. NN ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> E! a e. NN E! b e. NN ph )
6	1	2sqreunnlem2	|- A. a e. NN E* b e. NN ph
7		2reu1	|- ( A. a e. NN E* b e. NN ph -> ( E! a e. NN E! b e. NN ph <-> ( E! a e. NN E. b e. NN ph /\ E! b e. NN E. a e. NN ph ) ) )
8	6 7	ax-mp	|- ( E! a e. NN E! b e. NN ph <-> ( E! a e. NN E. b e. NN ph /\ E! b e. NN E. a e. NN ph ) )
9	5 8	bitri	|- ( E! a e. NN E! b e. NN ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> ( E! a e. NN E. b e. NN ph /\ E! b e. NN E. a e. NN ph ) )
10	2 9	bitrdi	|- ( P e. Prime -> ( ( P mod 4 ) = 1 <-> ( E! a e. NN E. b e. NN ph /\ E! b e. NN E. a e. NN ph ) ) )