Metamath Proof Explorer

Theorem elzs12

Description: Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025)

Ref Expression
Assertion elzs12 
|- ( A e. ZZ_s[1/2] <-> E. x e. ZZ_s E. y e. NN0_s A = ( x /su ( 2s ^su y ) ) )

Proof

Step Hyp Ref Expression
1 elex 
 |-  ( A e. ZZ_s[1/2] -> A e. _V )
2 id 
 |-  ( A = ( x /su ( 2s ^su y ) ) -> A = ( x /su ( 2s ^su y ) ) )
3 ovex 
 |-  ( x /su ( 2s ^su y ) ) e. _V
4 2 3 eqeltrdi 
 |-  ( A = ( x /su ( 2s ^su y ) ) -> A e. _V )
5 4 a1i 
 |-  ( ( x e. ZZ_s /\ y e. NN0_s ) -> ( A = ( x /su ( 2s ^su y ) ) -> A e. _V ) )
6 5 rexlimivv 
 |-  ( E. x e. ZZ_s E. y e. NN0_s A = ( x /su ( 2s ^su y ) ) -> A e. _V )
7 eqeq1 
 |-  ( z = A -> ( z = ( x /su ( 2s ^su y ) ) <-> A = ( x /su ( 2s ^su y ) ) ) )
8 7 2rexbidv 
 |-  ( z = A -> ( E. x e. ZZ_s E. y e. NN0_s z = ( x /su ( 2s ^su y ) ) <-> E. x e. ZZ_s E. y e. NN0_s A = ( x /su ( 2s ^su y ) ) ) )
9 df-zs12 
 |-  ZZ_s[1/2] = { z | E. x e. ZZ_s E. y e. NN0_s z = ( x /su ( 2s ^su y ) ) }
10 8 9 elab2g 
 |-  ( A e. _V -> ( A e. ZZ_s[1/2] <-> E. x e. ZZ_s E. y e. NN0_s A = ( x /su ( 2s ^su y ) ) ) )
11 1 6 10 pm5.21nii 
 |-  ( A e. ZZ_s[1/2] <-> E. x e. ZZ_s E. y e. NN0_s A = ( x /su ( 2s ^su y ) ) )