This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem absprodnn

Description: The absolute value of the product of the elements of a finite subset of the integers not containing 0 is a poitive integer. (Contributed by AV, 21-Aug-2020)

Ref Expression
Hypotheses absproddvds.s 
|- ( ph -> Z C_ ZZ )
absproddvds.f 
|- ( ph -> Z e. Fin )
absproddvds.p 
|- P = ( abs ` prod_ z e. Z z )
absprodnn.z 
|- ( ph -> 0 e/ Z )
Assertion absprodnn 
|- ( ph -> P e. NN )

Proof

Step Hyp Ref Expression
1 absproddvds.s 
 |-  ( ph -> Z C_ ZZ )
2 absproddvds.f 
 |-  ( ph -> Z e. Fin )
3 absproddvds.p 
 |-  P = ( abs ` prod_ z e. Z z )
4 absprodnn.z 
 |-  ( ph -> 0 e/ Z )
5 1 sselda 
 |-  ( ( ph /\ z e. Z ) -> z e. ZZ )
6 2 5 fprodzcl 
 |-  ( ph -> prod_ z e. Z z e. ZZ )
7 5 zcnd 
 |-  ( ( ph /\ z e. Z ) -> z e. CC )
8 elnelne2 
 |-  ( ( z e. Z /\ 0 e/ Z ) -> z =/= 0 )
9 8 expcom 
 |-  ( 0 e/ Z -> ( z e. Z -> z =/= 0 ) )
10 4 9 syl 
 |-  ( ph -> ( z e. Z -> z =/= 0 ) )
11 10 imp 
 |-  ( ( ph /\ z e. Z ) -> z =/= 0 )
12 2 7 11 fprodn0 
 |-  ( ph -> prod_ z e. Z z =/= 0 )
13 nnabscl 
 |-  ( ( prod_ z e. Z z e. ZZ /\ prod_ z e. Z z =/= 0 ) -> ( abs ` prod_ z e. Z z ) e. NN )
14 6 12 13 syl2anc 
 |-  ( ph -> ( abs ` prod_ z e. Z z ) e. NN )
15 3 14 eqeltrid 
 |-  ( ph -> P e. NN )