Step |
Hyp |
Ref |
Expression |
1 |
|
esummulc2.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
esummulc2.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
3 |
|
esummulc2.c |
⊢ ( 𝜑 → 𝐶 ∈ ( 0 [,) +∞ ) ) |
4 |
|
icossxr |
⊢ ( 0 [,) +∞ ) ⊆ ℝ* |
5 |
4 3
|
sselid |
⊢ ( 𝜑 → 𝐶 ∈ ℝ* ) |
6 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
7 |
2
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) |
8 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐴 |
9 |
8
|
esumcl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) |
10 |
1 7 9
|
syl2anc |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) |
11 |
6 10
|
sselid |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ℝ* ) |
12 |
|
xmulcom |
⊢ ( ( 𝐶 ∈ ℝ* ∧ Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ℝ* ) → ( 𝐶 ·e Σ* 𝑘 ∈ 𝐴 𝐵 ) = ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e 𝐶 ) ) |
13 |
5 11 12
|
syl2anc |
⊢ ( 𝜑 → ( 𝐶 ·e Σ* 𝑘 ∈ 𝐴 𝐵 ) = ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e 𝐶 ) ) |
14 |
1 2 3
|
esummulc1 |
⊢ ( 𝜑 → ( Σ* 𝑘 ∈ 𝐴 𝐵 ·e 𝐶 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐵 ·e 𝐶 ) ) |
15 |
6 2
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℝ* ) |
16 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℝ* ) |
17 |
|
xmulcom |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) ) |
18 |
15 16 17
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 ·e 𝐶 ) = ( 𝐶 ·e 𝐵 ) ) |
19 |
18
|
esumeq2dv |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 ( 𝐵 ·e 𝐶 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐶 ·e 𝐵 ) ) |
20 |
13 14 19
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐶 ·e Σ* 𝑘 ∈ 𝐴 𝐵 ) = Σ* 𝑘 ∈ 𝐴 ( 𝐶 ·e 𝐵 ) ) |