Theorem flmod

Description: The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008)

	Ref	Expression
Assertion	flmod	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) = ( 𝐴 − ( 𝐴 mod 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		modfrac	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 mod 1 ) = ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) )
2	1	oveq2d	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − ( 𝐴 mod 1 ) ) = ( 𝐴 − ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ) )
3		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
4		reflcl	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ )
5	4	recnd	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℂ )
6	3 5	nncand	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ) = ( ⌊ ‘ 𝐴 ) )
7	2 6	eqtr2d	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) = ( 𝐴 − ( 𝐴 mod 1 ) ) )