Metamath Proof Explorer

Theorem fznn0sub2

Description: Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion fznn0sub2 
|- ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) )

Proof

Step Hyp Ref Expression
1 elfzle1 
 |-  ( K e. ( 0 ... N ) -> 0 <_ K )
2 elfzel2 
 |-  ( K e. ( 0 ... N ) -> N e. ZZ )
3 elfzelz 
 |-  ( K e. ( 0 ... N ) -> K e. ZZ )
4 zre 
 |-  ( N e. ZZ -> N e. RR )
5 zre 
 |-  ( K e. ZZ -> K e. RR )
6 subge02 
 |-  ( ( N e. RR /\ K e. RR ) -> ( 0 <_ K <-> ( N - K ) <_ N ) )
7 4 5 6 syl2an 
 |-  ( ( N e. ZZ /\ K e. ZZ ) -> ( 0 <_ K <-> ( N - K ) <_ N ) )
8 2 3 7 syl2anc 
 |-  ( K e. ( 0 ... N ) -> ( 0 <_ K <-> ( N - K ) <_ N ) )
9 1 8 mpbid 
 |-  ( K e. ( 0 ... N ) -> ( N - K ) <_ N )
10 fznn0sub 
 |-  ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 )
11 nn0uz 
 |-  NN0 = ( ZZ>= ` 0 )
12 10 11 eleqtrdi 
 |-  ( K e. ( 0 ... N ) -> ( N - K ) e. ( ZZ>= ` 0 ) )
13 elfz5 
 |-  ( ( ( N - K ) e. ( ZZ>= ` 0 ) /\ N e. ZZ ) -> ( ( N - K ) e. ( 0 ... N ) <-> ( N - K ) <_ N ) )
14 12 2 13 syl2anc 
 |-  ( K e. ( 0 ... N ) -> ( ( N - K ) e. ( 0 ... N ) <-> ( N - K ) <_ N ) )
15 9 14 mpbird 
 |-  ( K e. ( 0 ... N ) -> ( N - K ) e. ( 0 ... N ) )