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

Metamath Proof Explorer

Theorem nnm1nn0

Description: A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007) (Revised by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nnm1nn0 
|- ( N e. NN -> ( N - 1 ) e. NN0 )

Proof

Step Hyp Ref Expression
1 nn1m1nn 
 |-  ( N e. NN -> ( N = 1 \/ ( N - 1 ) e. NN ) )
2 oveq1 
 |-  ( N = 1 -> ( N - 1 ) = ( 1 - 1 ) )
3 1m1e0 
 |-  ( 1 - 1 ) = 0
4 2 3 eqtrdi 
 |-  ( N = 1 -> ( N - 1 ) = 0 )
5 4 orim1i 
 |-  ( ( N = 1 \/ ( N - 1 ) e. NN ) -> ( ( N - 1 ) = 0 \/ ( N - 1 ) e. NN ) )
6 1 5 syl 
 |-  ( N e. NN -> ( ( N - 1 ) = 0 \/ ( N - 1 ) e. NN ) )
7 6 orcomd 
 |-  ( N e. NN -> ( ( N - 1 ) e. NN \/ ( N - 1 ) = 0 ) )
8 elnn0 
 |-  ( ( N - 1 ) e. NN0 <-> ( ( N - 1 ) e. NN \/ ( N - 1 ) = 0 ) )
9 7 8 sylibr 
 |-  ( N e. NN -> ( N - 1 ) e. NN0 )