Metamath Proof Explorer


Theorem xdivid

Description: A number divided by itself is one. (Contributed by Thierry Arnoux, 18-Dec-2016)

Ref Expression
Assertion xdivid ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 /𝑒 𝐴 ) = 1 )

Proof

Step Hyp Ref Expression
1 rexdiv ( ( 𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 /𝑒 𝐴 ) = ( 𝐴 / 𝐴 ) )
2 1 3anidm12 ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 /𝑒 𝐴 ) = ( 𝐴 / 𝐴 ) )
3 recn ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
4 divid ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = 1 )
5 3 4 sylan ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐴 ) = 1 )
6 2 5 eqtrd ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 /𝑒 𝐴 ) = 1 )