Metamath Proof Explorer


Theorem climprod1

Description: The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017)

Ref Expression
Hypotheses climprod1.1
|- Z = ( ZZ>= ` M )
climprod1.2
|- ( ph -> M e. ZZ )
Assertion climprod1
|- ( ph -> seq M ( x. , ( Z X. { 1 } ) ) ~~> 1 )

Proof

Step Hyp Ref Expression
1 climprod1.1
 |-  Z = ( ZZ>= ` M )
2 climprod1.2
 |-  ( ph -> M e. ZZ )
3 1 prodfclim1
 |-  ( M e. ZZ -> seq M ( x. , ( Z X. { 1 } ) ) ~~> 1 )
4 2 3 syl
 |-  ( ph -> seq M ( x. , ( Z X. { 1 } ) ) ~~> 1 )