Step |
Hyp |
Ref |
Expression |
1 |
|
flidm |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) = ( ⌊ ‘ 𝐴 ) ) |
2 |
1
|
oveq2d |
⊢ ( 𝐴 ∈ ℝ → ( 1 ... ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) ) = ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) |
3 |
2
|
sumeq1d |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) ) ( Λ ‘ 𝑥 ) = Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑥 ) ) |
4 |
|
reflcl |
⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) |
5 |
|
chpval |
⊢ ( ( ⌊ ‘ 𝐴 ) ∈ ℝ → ( ψ ‘ ( ⌊ ‘ 𝐴 ) ) = Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) ) ( Λ ‘ 𝑥 ) ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 ∈ ℝ → ( ψ ‘ ( ⌊ ‘ 𝐴 ) ) = Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) ) ( Λ ‘ 𝑥 ) ) |
7 |
|
chpval |
⊢ ( 𝐴 ∈ ℝ → ( ψ ‘ 𝐴 ) = Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑥 ) ) |
8 |
3 6 7
|
3eqtr4d |
⊢ ( 𝐴 ∈ ℝ → ( ψ ‘ ( ⌊ ‘ 𝐴 ) ) = ( ψ ‘ 𝐴 ) ) |