Metamath Proof Explorer


Theorem resubid

Description: Subtraction of a real number from itself (compare subid ). (Contributed by SN, 23-Jan-2024)

Ref Expression
Assertion resubid ( 𝐴 ∈ ℝ → ( 𝐴 𝐴 ) = 0 )

Proof

Step Hyp Ref Expression
1 re0m0e0 ( 0 − 0 ) = 0
2 1 oveq2i ( 𝐴 · ( 0 − 0 ) ) = ( 𝐴 · 0 )
3 sn-00idlem1 ( 𝐴 ∈ ℝ → ( 𝐴 · ( 0 − 0 ) ) = ( 𝐴 𝐴 ) )
4 remul01 ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = 0 )
5 2 3 4 3eqtr3a ( 𝐴 ∈ ℝ → ( 𝐴 𝐴 ) = 0 )