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Metamath Proof Explorer

Theorem xrs0

Description: The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass and df-xrs ), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017)

Ref Expression
Assertion xrs0 
|- 0 = ( 0g ` RR*s )

Proof

Step Hyp Ref Expression
1 xrsbas 
 |-  RR* = ( Base ` RR*s )
2 1 a1i 
 |-  ( T. -> RR* = ( Base ` RR*s ) )
3 xrsadd 
 |-  +e = ( +g ` RR*s )
4 3 a1i 
 |-  ( T. -> +e = ( +g ` RR*s ) )
5 0xr 
 |-  0 e. RR*
6 5 a1i 
 |-  ( T. -> 0 e. RR* )
7 xaddlid 
 |-  ( x e. RR* -> ( 0 +e x ) = x )
8 7 adantl 
 |-  ( ( T. /\ x e. RR* ) -> ( 0 +e x ) = x )
9 xaddrid 
 |-  ( x e. RR* -> ( x +e 0 ) = x )
10 9 adantl 
 |-  ( ( T. /\ x e. RR* ) -> ( x +e 0 ) = x )
11 2 4 6 8 10 grpidd 
 |-  ( T. -> 0 = ( 0g ` RR*s ) )
12 11 mptru 
 |-  0 = ( 0g ` RR*s )