Step |
Hyp |
Ref |
Expression |
1 |
|
1rp |
⊢ 1 ∈ ℝ+ |
2 |
|
modval |
⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ∈ ℝ+ ) → ( 𝐴 mod 1 ) = ( 𝐴 − ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) ) ) |
3 |
1 2
|
mpan2 |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 mod 1 ) = ( 𝐴 − ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) ) ) |
4 |
|
recn |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ ) |
5 |
4
|
div1d |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 / 1 ) = 𝐴 ) |
6 |
5
|
fveq2d |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ ( 𝐴 / 1 ) ) = ( ⌊ ‘ 𝐴 ) ) |
7 |
6
|
oveq2d |
⊢ ( 𝐴 ∈ ℝ → ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) = ( 1 · ( ⌊ ‘ 𝐴 ) ) ) |
8 |
|
reflcl |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) |
9 |
8
|
recnd |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℂ ) |
10 |
9
|
mulid2d |
⊢ ( 𝐴 ∈ ℝ → ( 1 · ( ⌊ ‘ 𝐴 ) ) = ( ⌊ ‘ 𝐴 ) ) |
11 |
7 10
|
eqtrd |
⊢ ( 𝐴 ∈ ℝ → ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) = ( ⌊ ‘ 𝐴 ) ) |
12 |
11
|
oveq2d |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) ) = ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ) |
13 |
3 12
|
eqtrd |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 mod 1 ) = ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ) |