Metamath Proof Explorer


Theorem xaddid2

Description: Extended real version of addid2 . (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xaddid2 ( 𝐴 ∈ ℝ* → ( 0 +𝑒 𝐴 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 0xr 0 ∈ ℝ*
2 xaddcom ( ( 0 ∈ ℝ*𝐴 ∈ ℝ* ) → ( 0 +𝑒 𝐴 ) = ( 𝐴 +𝑒 0 ) )
3 1 2 mpan ( 𝐴 ∈ ℝ* → ( 0 +𝑒 𝐴 ) = ( 𝐴 +𝑒 0 ) )
4 xaddid1 ( 𝐴 ∈ ℝ* → ( 𝐴 +𝑒 0 ) = 𝐴 )
5 3 4 eqtrd ( 𝐴 ∈ ℝ* → ( 0 +𝑒 𝐴 ) = 𝐴 )