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Metamath Proof Explorer

Theorem seqabs

Description: Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values. (Contributed by Mario Carneiro, 26-Mar-2014) (Revised by Mario Carneiro, 27-May-2014)

Ref Expression
Hypotheses seqabs.1 
|- ( ph -> N e. ( ZZ>= ` M ) )
seqabs.2 
|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
seqabs.3 
|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
Assertion seqabs 
|- ( ph -> ( abs ` ( seq M ( + , F ) ` N ) ) <_ ( seq M ( + , G ) ` N ) )

Proof

Step Hyp Ref Expression
1 seqabs.1 
 |-  ( ph -> N e. ( ZZ>= ` M ) )
2 seqabs.2 
 |-  ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
3 seqabs.3 
 |-  ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
4 fzfid 
 |-  ( ph -> ( M ... N ) e. Fin )
5 4 2 fsumabs 
 |-  ( ph -> ( abs ` sum_ k e. ( M ... N ) ( F ` k ) ) <_ sum_ k e. ( M ... N ) ( abs ` ( F ` k ) ) )
6 eqidd 
 |-  ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( F ` k ) )
7 6 1 2 fsumser 
 |-  ( ph -> sum_ k e. ( M ... N ) ( F ` k ) = ( seq M ( + , F ) ` N ) )
8 7 fveq2d 
 |-  ( ph -> ( abs ` sum_ k e. ( M ... N ) ( F ` k ) ) = ( abs ` ( seq M ( + , F ) ` N ) ) )
9 abscl 
 |-  ( ( F ` k ) e. CC -> ( abs ` ( F ` k ) ) e. RR )
10 9 recnd 
 |-  ( ( F ` k ) e. CC -> ( abs ` ( F ` k ) ) e. CC )
11 2 10 syl 
 |-  ( ( ph /\ k e. ( M ... N ) ) -> ( abs ` ( F ` k ) ) e. CC )
12 3 1 11 fsumser 
 |-  ( ph -> sum_ k e. ( M ... N ) ( abs ` ( F ` k ) ) = ( seq M ( + , G ) ` N ) )
13 5 8 12 3brtr3d 
 |-  ( ph -> ( abs ` ( seq M ( + , F ) ` N ) ) <_ ( seq M ( + , G ) ` N ) )