Step |
Hyp |
Ref |
Expression |
1 |
|
fladdz |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ) → ( ⌊ ‘ ( 𝐴 + 𝑁 ) ) = ( ( ⌊ ‘ 𝐴 ) + 𝑁 ) ) |
2 |
|
recn |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ ) |
3 |
|
zcn |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℂ ) |
4 |
|
addcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ ) → ( 𝐴 + 𝑁 ) = ( 𝑁 + 𝐴 ) ) |
5 |
2 3 4
|
syl2an |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 + 𝑁 ) = ( 𝑁 + 𝐴 ) ) |
6 |
5
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ) → ( ⌊ ‘ ( 𝐴 + 𝑁 ) ) = ( ⌊ ‘ ( 𝑁 + 𝐴 ) ) ) |
7 |
|
reflcl |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) |
8 |
7
|
recnd |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℂ ) |
9 |
|
addcom |
⊢ ( ( ( ⌊ ‘ 𝐴 ) ∈ ℂ ∧ 𝑁 ∈ ℂ ) → ( ( ⌊ ‘ 𝐴 ) + 𝑁 ) = ( 𝑁 + ( ⌊ ‘ 𝐴 ) ) ) |
10 |
8 3 9
|
syl2an |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ) → ( ( ⌊ ‘ 𝐴 ) + 𝑁 ) = ( 𝑁 + ( ⌊ ‘ 𝐴 ) ) ) |
11 |
1 6 10
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ) → ( ⌊ ‘ ( 𝑁 + 𝐴 ) ) = ( 𝑁 + ( ⌊ ‘ 𝐴 ) ) ) |
12 |
11
|
ancoms |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℝ ) → ( ⌊ ‘ ( 𝑁 + 𝐴 ) ) = ( 𝑁 + ( ⌊ ‘ 𝐴 ) ) ) |