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Metamath Proof Explorer

Theorem zzlesq

Description: An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025)

Ref Expression
Assertion zzlesq 
|- ( N e. ZZ -> N <_ ( N ^ 2 ) )

Proof

Step Hyp Ref Expression
1 elznn 
 |-  ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) )
2 animorrl 
 |-  ( ( N e. RR /\ N e. NN ) -> ( N e. NN \/ ( N e. RR /\ -u N e. NN0 ) ) )
3 olc 
 |-  ( ( N e. RR /\ -u N e. NN0 ) -> ( N e. NN \/ ( N e. RR /\ -u N e. NN0 ) ) )
4 2 3 jaodan 
 |-  ( ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) -> ( N e. NN \/ ( N e. RR /\ -u N e. NN0 ) ) )
5 1 4 sylbi 
 |-  ( N e. ZZ -> ( N e. NN \/ ( N e. RR /\ -u N e. NN0 ) ) )
6 nnlesq 
 |-  ( N e. NN -> N <_ ( N ^ 2 ) )
7 simpl 
 |-  ( ( N e. RR /\ -u N e. NN0 ) -> N e. RR )
8 0red 
 |-  ( ( N e. RR /\ -u N e. NN0 ) -> 0 e. RR )
9 7 resqcld 
 |-  ( ( N e. RR /\ -u N e. NN0 ) -> ( N ^ 2 ) e. RR )
10 nn0ge0 
 |-  ( -u N e. NN0 -> 0 <_ -u N )
11 le0neg1 
 |-  ( N e. RR -> ( N <_ 0 <-> 0 <_ -u N ) )
12 11 biimpar 
 |-  ( ( N e. RR /\ 0 <_ -u N ) -> N <_ 0 )
13 10 12 sylan2 
 |-  ( ( N e. RR /\ -u N e. NN0 ) -> N <_ 0 )
14 7 sqge0d 
 |-  ( ( N e. RR /\ -u N e. NN0 ) -> 0 <_ ( N ^ 2 ) )
15 7 8 9 13 14 letrd 
 |-  ( ( N e. RR /\ -u N e. NN0 ) -> N <_ ( N ^ 2 ) )
16 6 15 jaoi 
 |-  ( ( N e. NN \/ ( N e. RR /\ -u N e. NN0 ) ) -> N <_ ( N ^ 2 ) )
17 5 16 syl 
 |-  ( N e. ZZ -> N <_ ( N ^ 2 ) )