Step |
Hyp |
Ref |
Expression |
1 |
|
sge0ss.kph |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
sge0ss.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑉 ) |
3 |
|
sge0ss.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
4 |
|
sge0ss.c |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
5 |
|
sge0ss.c0 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ) → 𝐶 = 0 ) |
6 |
|
ssexg |
⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ V ) |
7 |
3 2 6
|
syl2anc |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
8 |
2
|
difexd |
⊢ ( 𝜑 → ( 𝐵 ∖ 𝐴 ) ∈ V ) |
9 |
|
disjdif |
⊢ ( 𝐴 ∩ ( 𝐵 ∖ 𝐴 ) ) = ∅ |
10 |
9
|
a1i |
⊢ ( 𝜑 → ( 𝐴 ∩ ( 𝐵 ∖ 𝐴 ) ) = ∅ ) |
11 |
|
0e0iccpnf |
⊢ 0 ∈ ( 0 [,] +∞ ) |
12 |
11
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ) → 0 ∈ ( 0 [,] +∞ ) ) |
13 |
5 12
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
14 |
1 7 8 10 4 13
|
sge0splitmpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) ↦ 𝐶 ) ) = ( ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) +𝑒 ( Σ^ ‘ ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 𝐶 ) ) ) ) |
15 |
14
|
eqcomd |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) +𝑒 ( Σ^ ‘ ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 𝐶 ) ) ) = ( Σ^ ‘ ( 𝑘 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) ↦ 𝐶 ) ) ) |
16 |
1 5
|
mpteq2da |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 𝐶 ) = ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 0 ) ) |
17 |
16
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 𝐶 ) ) = ( Σ^ ‘ ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 0 ) ) ) |
18 |
1 8
|
sge0z |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 0 ) ) = 0 ) |
19 |
17 18
|
eqtrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 𝐶 ) ) = 0 ) |
20 |
19
|
oveq2d |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) +𝑒 ( Σ^ ‘ ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 𝐶 ) ) ) = ( ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) +𝑒 0 ) ) |
21 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) |
22 |
1 4 21
|
fmptdf |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
23 |
7 22
|
sge0xrcl |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) ∈ ℝ* ) |
24 |
|
xaddid1 |
⊢ ( ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) ∈ ℝ* → ( ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) +𝑒 0 ) = ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → ( ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) +𝑒 0 ) = ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) ) |
26 |
|
eqidd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) = ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) ) |
27 |
20 25 26
|
3eqtrrd |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) = ( ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) +𝑒 ( Σ^ ‘ ( 𝑘 ∈ ( 𝐵 ∖ 𝐴 ) ↦ 𝐶 ) ) ) ) |
28 |
|
undif |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 ) |
29 |
3 28
|
sylib |
⊢ ( 𝜑 → ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 ) |
30 |
29
|
eqcomd |
⊢ ( 𝜑 → 𝐵 = ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) ) |
31 |
30
|
mpteq1d |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐵 ↦ 𝐶 ) = ( 𝑘 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) ↦ 𝐶 ) ) |
32 |
31
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ 𝐵 ↦ 𝐶 ) ) = ( Σ^ ‘ ( 𝑘 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) ↦ 𝐶 ) ) ) |
33 |
15 27 32
|
3eqtr4d |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ) = ( Σ^ ‘ ( 𝑘 ∈ 𝐵 ↦ 𝐶 ) ) ) |