Step |
Hyp |
Ref |
Expression |
1 |
|
sge0prle.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
sge0prle.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
|
sge0prle.d |
⊢ ( 𝜑 → 𝐷 ∈ ( 0 [,] +∞ ) ) |
4 |
|
sge0prle.e |
⊢ ( 𝜑 → 𝐸 ∈ ( 0 [,] +∞ ) ) |
5 |
|
sge0prle.cd |
⊢ ( 𝑘 = 𝐴 → 𝐶 = 𝐷 ) |
6 |
|
sge0prle.ce |
⊢ ( 𝑘 = 𝐵 → 𝐶 = 𝐸 ) |
7 |
|
preq1 |
⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐵 } = { 𝐵 , 𝐵 } ) |
8 |
|
dfsn2 |
⊢ { 𝐵 } = { 𝐵 , 𝐵 } |
9 |
8
|
eqcomi |
⊢ { 𝐵 , 𝐵 } = { 𝐵 } |
10 |
9
|
a1i |
⊢ ( 𝐴 = 𝐵 → { 𝐵 , 𝐵 } = { 𝐵 } ) |
11 |
7 10
|
eqtrd |
⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐵 } = { 𝐵 } ) |
12 |
11
|
mpteq1d |
⊢ ( 𝐴 = 𝐵 → ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) = ( 𝑘 ∈ { 𝐵 } ↦ 𝐶 ) ) |
13 |
12
|
fveq2d |
⊢ ( 𝐴 = 𝐵 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( Σ^ ‘ ( 𝑘 ∈ { 𝐵 } ↦ 𝐶 ) ) ) |
14 |
13
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( Σ^ ‘ ( 𝑘 ∈ { 𝐵 } ↦ 𝐶 ) ) ) |
15 |
2 4 6
|
sge0snmpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐵 } ↦ 𝐶 ) ) = 𝐸 ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐵 } ↦ 𝐶 ) ) = 𝐸 ) |
17 |
14 16
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = 𝐸 ) |
18 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
19 |
18 4
|
sselid |
⊢ ( 𝜑 → 𝐸 ∈ ℝ* ) |
20 |
19
|
xaddid2d |
⊢ ( 𝜑 → ( 0 +𝑒 𝐸 ) = 𝐸 ) |
21 |
20
|
eqcomd |
⊢ ( 𝜑 → 𝐸 = ( 0 +𝑒 𝐸 ) ) |
22 |
|
0xr |
⊢ 0 ∈ ℝ* |
23 |
22
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ* ) |
24 |
18 3
|
sselid |
⊢ ( 𝜑 → 𝐷 ∈ ℝ* ) |
25 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
26 |
25
|
a1i |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
27 |
|
iccgelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐷 ∈ ( 0 [,] +∞ ) ) → 0 ≤ 𝐷 ) |
28 |
23 26 3 27
|
syl3anc |
⊢ ( 𝜑 → 0 ≤ 𝐷 ) |
29 |
23 24 19 28
|
xleadd1d |
⊢ ( 𝜑 → ( 0 +𝑒 𝐸 ) ≤ ( 𝐷 +𝑒 𝐸 ) ) |
30 |
21 29
|
eqbrtrd |
⊢ ( 𝜑 → 𝐸 ≤ ( 𝐷 +𝑒 𝐸 ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐸 ≤ ( 𝐷 +𝑒 𝐸 ) ) |
32 |
17 31
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) ≤ ( 𝐷 +𝑒 𝐸 ) ) |
33 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝑉 ) |
34 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝑊 ) |
35 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐷 ∈ ( 0 [,] +∞ ) ) |
36 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐸 ∈ ( 0 [,] +∞ ) ) |
37 |
|
neqne |
⊢ ( ¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵 ) |
38 |
37
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → 𝐴 ≠ 𝐵 ) |
39 |
33 34 35 36 5 6 38
|
sge0pr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) = ( 𝐷 +𝑒 𝐸 ) ) |
40 |
24 19
|
xaddcld |
⊢ ( 𝜑 → ( 𝐷 +𝑒 𝐸 ) ∈ ℝ* ) |
41 |
40
|
xrleidd |
⊢ ( 𝜑 → ( 𝐷 +𝑒 𝐸 ) ≤ ( 𝐷 +𝑒 𝐸 ) ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ( 𝐷 +𝑒 𝐸 ) ≤ ( 𝐷 +𝑒 𝐸 ) ) |
43 |
39 42
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 = 𝐵 ) → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) ≤ ( 𝐷 +𝑒 𝐸 ) ) |
44 |
32 43
|
pm2.61dan |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ { 𝐴 , 𝐵 } ↦ 𝐶 ) ) ≤ ( 𝐷 +𝑒 𝐸 ) ) |