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 )