Theorem fsumfldivdiag

Description: The right-hand side of dvdsflsumcom is commutative in the variables, because it can be written as the manifestly symmetric sum over those <. m , n >. such that m x. n <_ A . (Contributed by Mario Carneiro, 4-May-2016)

	Ref	Expression
Hypotheses	fsumfldivdiag.1	|- ( ph -> A e. RR )
	fsumfldivdiag.2	|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> B e. CC )
Assertion	fsumfldivdiag	|- ( ph -> sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ m e. ( 1 ... ( |_ ` ( A / n ) ) ) B = sum_ m e. ( 1 ... ( |_ ` A ) ) sum_ n e. ( 1 ... ( |_ ` ( A / m ) ) ) B )

Proof

Step	Hyp	Ref	Expression
1		fsumfldivdiag.1	|- ( ph -> A e. RR )
2		fsumfldivdiag.2	|- ( ( ph /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) -> B e. CC )
3		fzfid	|- ( ph -> ( 1 ... ( |_ ` A ) ) e. Fin )
4		fzfid	|- ( ( ph /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( 1 ... ( |_ ` ( A / n ) ) ) e. Fin )
5	1	fsumfldivdiaglem	|- ( ph -> ( ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) -> ( m e. ( 1 ... ( |_ ` A ) ) /\ n e. ( 1 ... ( |_ ` ( A / m ) ) ) ) ) )
6	1	fsumfldivdiaglem	|- ( ph -> ( ( m e. ( 1 ... ( |_ ` A ) ) /\ n e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) ) )
7	5 6	impbid	|- ( ph -> ( ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. ( 1 ... ( |_ ` ( A / n ) ) ) ) <-> ( m e. ( 1 ... ( |_ ` A ) ) /\ n e. ( 1 ... ( |_ ` ( A / m ) ) ) ) ) )
8	3 3 4 7 2	fsumcom2	|- ( ph -> sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ m e. ( 1 ... ( |_ ` ( A / n ) ) ) B = sum_ m e. ( 1 ... ( |_ ` A ) ) sum_ n e. ( 1 ... ( |_ ` ( A / m ) ) ) B )