Metamath Proof Explorer


Theorem renegneg

Description: A real number is equal to the negative of its negative. Compare negneg . (Contributed by SN, 13-Feb-2024)

Ref Expression
Assertion renegneg
|- ( A e. RR -> ( 0 -R ( 0 -R A ) ) = A )

Proof

Step Hyp Ref Expression
1 rernegcl
 |-  ( A e. RR -> ( 0 -R A ) e. RR )
2 rernegcl
 |-  ( ( 0 -R A ) e. RR -> ( 0 -R ( 0 -R A ) ) e. RR )
3 1 2 syl
 |-  ( A e. RR -> ( 0 -R ( 0 -R A ) ) e. RR )
4 id
 |-  ( A e. RR -> A e. RR )
5 renegid
 |-  ( A e. RR -> ( A + ( 0 -R A ) ) = 0 )
6 elre0re
 |-  ( A e. RR -> 0 e. RR )
7 5 6 eqeltrd
 |-  ( A e. RR -> ( A + ( 0 -R A ) ) e. RR )
8 readdid1
 |-  ( A e. RR -> ( A + 0 ) = A )
9 repncan3
 |-  ( ( ( 0 -R A ) e. RR /\ 0 e. RR ) -> ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) = 0 )
10 1 6 9 syl2anc
 |-  ( A e. RR -> ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) = 0 )
11 10 oveq2d
 |-  ( A e. RR -> ( A + ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) ) = ( A + 0 ) )
12 readdid2
 |-  ( A e. RR -> ( 0 + A ) = A )
13 8 11 12 3eqtr4d
 |-  ( A e. RR -> ( A + ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) ) = ( 0 + A ) )
14 recn
 |-  ( A e. RR -> A e. CC )
15 1 recnd
 |-  ( A e. RR -> ( 0 -R A ) e. CC )
16 3 recnd
 |-  ( A e. RR -> ( 0 -R ( 0 -R A ) ) e. CC )
17 14 15 16 addassd
 |-  ( A e. RR -> ( ( A + ( 0 -R A ) ) + ( 0 -R ( 0 -R A ) ) ) = ( A + ( ( 0 -R A ) + ( 0 -R ( 0 -R A ) ) ) ) )
18 5 oveq1d
 |-  ( A e. RR -> ( ( A + ( 0 -R A ) ) + A ) = ( 0 + A ) )
19 13 17 18 3eqtr4d
 |-  ( A e. RR -> ( ( A + ( 0 -R A ) ) + ( 0 -R ( 0 -R A ) ) ) = ( ( A + ( 0 -R A ) ) + A ) )
20 readdcan
 |-  ( ( ( 0 -R ( 0 -R A ) ) e. RR /\ A e. RR /\ ( A + ( 0 -R A ) ) e. RR ) -> ( ( ( A + ( 0 -R A ) ) + ( 0 -R ( 0 -R A ) ) ) = ( ( A + ( 0 -R A ) ) + A ) <-> ( 0 -R ( 0 -R A ) ) = A ) )
21 20 biimpa
 |-  ( ( ( ( 0 -R ( 0 -R A ) ) e. RR /\ A e. RR /\ ( A + ( 0 -R A ) ) e. RR ) /\ ( ( A + ( 0 -R A ) ) + ( 0 -R ( 0 -R A ) ) ) = ( ( A + ( 0 -R A ) ) + A ) ) -> ( 0 -R ( 0 -R A ) ) = A )
22 3 4 7 19 21 syl31anc
 |-  ( A e. RR -> ( 0 -R ( 0 -R A ) ) = A )