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Metamath Proof Explorer

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 )