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Metamath Proof Explorer

Theorem elfzouz2

Description: The upper bound of a half-open range is greater than or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015)

Ref Expression
Assertion elfzouz2 
|- ( K e. ( M ..^ N ) -> N e. ( ZZ>= ` K ) )

Proof

Step Hyp Ref Expression
1 elfzoelz 
 |-  ( K e. ( M ..^ N ) -> K e. ZZ )
2 elfzoel2 
 |-  ( K e. ( M ..^ N ) -> N e. ZZ )
3 elfzolt2 
 |-  ( K e. ( M ..^ N ) -> K < N )
4 zre 
 |-  ( K e. ZZ -> K e. RR )
5 zre 
 |-  ( N e. ZZ -> N e. RR )
6 ltle 
 |-  ( ( K e. RR /\ N e. RR ) -> ( K < N -> K <_ N ) )
7 4 5 6 syl2an 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( K < N -> K <_ N ) )
8 1 2 7 syl2anc 
 |-  ( K e. ( M ..^ N ) -> ( K < N -> K <_ N ) )
9 3 8 mpd 
 |-  ( K e. ( M ..^ N ) -> K <_ N )
10 eluz2 
 |-  ( N e. ( ZZ>= ` K ) <-> ( K e. ZZ /\ N e. ZZ /\ K <_ N ) )
11 1 2 9 10 syl3anbrc 
 |-  ( K e. ( M ..^ N ) -> N e. ( ZZ>= ` K ) )