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

Metamath Proof Explorer

Theorem nn0sqeq1

Description: A natural number with square one is equal to one. (Contributed by Thierry Arnoux, 2-Feb-2020) (Proof shortened by Thierry Arnoux, 6-Jun-2023)

Ref Expression
Assertion nn0sqeq1 
|- ( ( N e. NN0 /\ ( N ^ 2 ) = 1 ) -> N = 1 )

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( N e. NN0 /\ ( N ^ 2 ) = 1 ) -> ( N ^ 2 ) = 1 )
2 1 fveq2d 
 |-  ( ( N e. NN0 /\ ( N ^ 2 ) = 1 ) -> ( sqrt ` ( N ^ 2 ) ) = ( sqrt ` 1 ) )
3 nn0re 
 |-  ( N e. NN0 -> N e. RR )
4 nn0ge0 
 |-  ( N e. NN0 -> 0 <_ N )
5 sqrtsq 
 |-  ( ( N e. RR /\ 0 <_ N ) -> ( sqrt ` ( N ^ 2 ) ) = N )
6 3 4 5 syl2anc 
 |-  ( N e. NN0 -> ( sqrt ` ( N ^ 2 ) ) = N )
7 6 adantr 
 |-  ( ( N e. NN0 /\ ( N ^ 2 ) = 1 ) -> ( sqrt ` ( N ^ 2 ) ) = N )
8 sqrt1 
 |-  ( sqrt ` 1 ) = 1
9 8 a1i 
 |-  ( ( N e. NN0 /\ ( N ^ 2 ) = 1 ) -> ( sqrt ` 1 ) = 1 )
10 2 7 9 3eqtr3d 
 |-  ( ( N e. NN0 /\ ( N ^ 2 ) = 1 ) -> N = 1 )