Theorem intfrac

Description: Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008)

	Ref	Expression
Assertion	intfrac	⊢ ( 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		reflcl	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ )
4	3	recnd	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℂ )
5		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
6	4 5	pncan3d	⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) + ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ) = 𝐴 )
7	2 6	eqtr2d	⊢ ( 𝐴 ∈ ℝ → 𝐴 = ( ( ⌊ ‘ 𝐴 ) + ( 𝐴 mod 1 ) ) )