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Metamath Proof Explorer

Theorem uzneg

Description: Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005)

Ref Expression
Assertion uzneg 
|- ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) )

Proof

Step Hyp Ref Expression
1 eluzle 
 |-  ( N e. ( ZZ>= ` M ) -> M <_ N )
2 eluzel2 
 |-  ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3 eluzelz 
 |-  ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4 zre 
 |-  ( M e. ZZ -> M e. RR )
5 zre 
 |-  ( N e. ZZ -> N e. RR )
6 leneg 
 |-  ( ( M e. RR /\ N e. RR ) -> ( M <_ N <-> -u N <_ -u M ) )
7 4 5 6 syl2an 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> -u N <_ -u M ) )
8 2 3 7 syl2anc 
 |-  ( N e. ( ZZ>= ` M ) -> ( M <_ N <-> -u N <_ -u M ) )
9 1 8 mpbid 
 |-  ( N e. ( ZZ>= ` M ) -> -u N <_ -u M )
10 znegcl 
 |-  ( N e. ZZ -> -u N e. ZZ )
11 znegcl 
 |-  ( M e. ZZ -> -u M e. ZZ )
12 eluz 
 |-  ( ( -u N e. ZZ /\ -u M e. ZZ ) -> ( -u M e. ( ZZ>= ` -u N ) <-> -u N <_ -u M ) )
13 10 11 12 syl2an 
 |-  ( ( N e. ZZ /\ M e. ZZ ) -> ( -u M e. ( ZZ>= ` -u N ) <-> -u N <_ -u M ) )
14 3 2 13 syl2anc 
 |-  ( N e. ( ZZ>= ` M ) -> ( -u M e. ( ZZ>= ` -u N ) <-> -u N <_ -u M ) )
15 9 14 mpbird 
 |-  ( N e. ( ZZ>= ` M ) -> -u M e. ( ZZ>= ` -u N ) )