Proof of Theorem ovnsubadd2lem
Step | Hyp | Ref
| Expression |
1 | | ovnsubadd2lem.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ Fin) |
2 | | iftrue 4125 |
. . . . . . . 8
⊢ (𝑛 = 1 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴) |
3 | 2 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐴) |
4 | | ovexd 6720 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ
↑𝑚 𝑋) ∈ V) |
5 | | ovnsubadd2lem.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚
𝑋)) |
6 | 4, 5 | ssexd 4838 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ V) |
7 | 6, 5 | elpwd 4200 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
8 | 7 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
9 | 3, 8 | eqeltrd 2730 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
10 | 9 | adantlr 751 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
11 | | simpl 472 |
. . . . . . . . . . 11
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1) |
12 | 11 | iffalsed 4130 |
. . . . . . . . . 10
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅)) |
13 | | simpr 476 |
. . . . . . . . . . 11
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2) |
14 | 13 | iftrued 4127 |
. . . . . . . . . 10
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵) |
15 | 12, 14 | eqtrd 2685 |
. . . . . . . . 9
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵) |
16 | 15 | adantll 750 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵) |
17 | | ovnsubadd2lem.b |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑𝑚
𝑋)) |
18 | 4, 17 | ssexd 4838 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ V) |
19 | 18, 17 | elpwd 4200 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
20 | 19 | ad2antrr 762 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
21 | 16, 20 | eqeltrd 2730 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
22 | 21 | adantllr 755 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
23 | | simpl 472 |
. . . . . . . . . 10
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1) |
24 | 23 | iffalsed 4130 |
. . . . . . . . 9
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅)) |
25 | | simpr 476 |
. . . . . . . . . 10
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2) |
26 | 25 | iffalsed 4130 |
. . . . . . . . 9
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅) |
27 | 24, 26 | eqtrd 2685 |
. . . . . . . 8
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅) |
28 | | 0elpw 4864 |
. . . . . . . . 9
⊢ ∅
∈ 𝒫 (ℝ ↑𝑚 𝑋) |
29 | 28 | a1i 11 |
. . . . . . . 8
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈
𝒫 (ℝ ↑𝑚 𝑋)) |
30 | 27, 29 | eqeltrd 2730 |
. . . . . . 7
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
31 | 30 | adantll 750 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
32 | 22, 31 | pm2.61dan 849 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
33 | 10, 32 | pm2.61dan 849 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
34 | | ovnsubadd2lem.c |
. . . 4
⊢ 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅))) |
35 | 33, 34 | fmptd 6425 |
. . 3
⊢ (𝜑 → 𝐶:ℕ⟶𝒫 (ℝ
↑𝑚 𝑋)) |
36 | 1, 35 | ovnsubadd 41107 |
. 2
⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛))))) |
37 | | eldifi 3765 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ 𝑛 ∈
ℕ) |
38 | 37 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
𝑛 ∈
ℕ) |
39 | | eldifn 3766 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ ¬ 𝑛 ∈ {1,
2}) |
40 | | vex 3234 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑛 ∈ V |
41 | 40 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑛 ∈ {1, 2} →
𝑛 ∈
V) |
42 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑛 ∈ {1, 2} →
¬ 𝑛 ∈ {1,
2}) |
43 | 41, 42 | nelpr1 39576 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑛 ∈ {1, 2} →
𝑛 ≠ 1) |
44 | 43 | neneqd 2828 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑛 ∈ {1, 2} →
¬ 𝑛 =
1) |
45 | 39, 44 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ ¬ 𝑛 =
1) |
46 | 41, 42 | nelpr2 39575 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑛 ∈ {1, 2} →
𝑛 ≠ 2) |
47 | 46 | neneqd 2828 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑛 ∈ {1, 2} →
¬ 𝑛 =
2) |
48 | 39, 47 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ ¬ 𝑛 =
2) |
49 | 45, 48, 27 | syl2anc 694 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅) |
50 | | 0ex 4823 |
. . . . . . . . . . . . 13
⊢ ∅
∈ V |
51 | 50 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ ∅ ∈ V) |
52 | 49, 51 | eqeltrd 2730 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) |
53 | 52 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) |
54 | 34 | fvmpt2 6330 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ ℕ ∧ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ V) → (𝐶‘𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅))) |
55 | 38, 53, 54 | syl2anc 694 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
(𝐶‘𝑛) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅))) |
56 | 49 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅) |
57 | 55, 56 | eqtrd 2685 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
(𝐶‘𝑛) = ∅) |
58 | 57 | ralrimiva 2995 |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅) |
59 | | nfcv 2793 |
. . . . . . . 8
⊢
Ⅎ𝑛(ℕ ∖ {1, 2}) |
60 | 59 | iunxdif3 4638 |
. . . . . . 7
⊢
(∀𝑛 ∈
(ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅ → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1,
2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) |
61 | 58, 60 | syl 17 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1,
2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) |
62 | 61 | eqcomd 2657 |
. . . . 5
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖
(ℕ ∖ {1, 2}))(𝐶‘𝑛)) |
63 | | 1nn 11069 |
. . . . . . . . . 10
⊢ 1 ∈
ℕ |
64 | | 2nn 11223 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ |
65 | 63, 64 | pm3.2i 470 |
. . . . . . . . 9
⊢ (1 ∈
ℕ ∧ 2 ∈ ℕ) |
66 | | prssi 4385 |
. . . . . . . . 9
⊢ ((1
∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆
ℕ) |
67 | 65, 66 | ax-mp 5 |
. . . . . . . 8
⊢ {1, 2}
⊆ ℕ |
68 | | dfss4 3891 |
. . . . . . . 8
⊢ ({1, 2}
⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1,
2}) |
69 | 67, 68 | mpbi 220 |
. . . . . . 7
⊢ (ℕ
∖ (ℕ ∖ {1, 2})) = {1, 2} |
70 | | iuneq1 4566 |
. . . . . . 7
⊢ ((ℕ
∖ (ℕ ∖ {1, 2})) = {1, 2} → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1,
2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛)) |
71 | 69, 70 | ax-mp 5 |
. . . . . 6
⊢ ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1,
2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) |
72 | 71 | a1i 11 |
. . . . 5
⊢ (𝜑 → ∪ 𝑛 ∈ (ℕ ∖ (ℕ ∖ {1,
2}))(𝐶‘𝑛) = ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛)) |
73 | | fveq2 6229 |
. . . . . . . . 9
⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1)) |
74 | | fveq2 6229 |
. . . . . . . . 9
⊢ (𝑛 = 2 → (𝐶‘𝑛) = (𝐶‘2)) |
75 | 73, 74 | iunxprg 4639 |
. . . . . . . 8
⊢ ((1
∈ ℕ ∧ 2 ∈ ℕ) → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2))) |
76 | 63, 64, 75 | mp2an 708 |
. . . . . . 7
⊢ ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2)) |
77 | 76 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2))) |
78 | 34 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))) |
79 | 63 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℕ) |
80 | 78, 3, 79, 6 | fvmptd 6327 |
. . . . . . 7
⊢ (𝜑 → (𝐶‘1) = 𝐴) |
81 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 2 → 𝑛 = 2) |
82 | | 1ne2 11278 |
. . . . . . . . . . . . . . 15
⊢ 1 ≠
2 |
83 | 82 | necomi 2877 |
. . . . . . . . . . . . . 14
⊢ 2 ≠
1 |
84 | 83 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 2 → 2 ≠
1) |
85 | 81, 84 | eqnetrd 2890 |
. . . . . . . . . . . 12
⊢ (𝑛 = 2 → 𝑛 ≠ 1) |
86 | 85 | neneqd 2828 |
. . . . . . . . . . 11
⊢ (𝑛 = 2 → ¬ 𝑛 = 1) |
87 | 86 | iffalsed 4130 |
. . . . . . . . . 10
⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅)) |
88 | | iftrue 4125 |
. . . . . . . . . 10
⊢ (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵) |
89 | 87, 88 | eqtrd 2685 |
. . . . . . . . 9
⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵) |
90 | 89 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵) |
91 | 64 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 2 ∈
ℕ) |
92 | 78, 90, 91, 18 | fvmptd 6327 |
. . . . . . 7
⊢ (𝜑 → (𝐶‘2) = 𝐵) |
93 | 80, 92 | uneq12d 3801 |
. . . . . 6
⊢ (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴 ∪ 𝐵)) |
94 | | eqidd 2652 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)) |
95 | 77, 93, 94 | 3eqtrd 2689 |
. . . . 5
⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = (𝐴 ∪ 𝐵)) |
96 | 62, 72, 95 | 3eqtrd 2689 |
. . . 4
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = (𝐴 ∪ 𝐵)) |
97 | 96 | fveq2d 6233 |
. . 3
⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐴 ∪ 𝐵))) |
98 | | nfv 1883 |
. . . . . 6
⊢
Ⅎ𝑛𝜑 |
99 | | nnex 11064 |
. . . . . . 7
⊢ ℕ
∈ V |
100 | 99 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℕ ∈
V) |
101 | 67 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {1, 2} ⊆
ℕ) |
102 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin) |
103 | | simpl 472 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝜑) |
104 | 101 | sselda 3636 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ) |
105 | 35 | ffvelrnda 6399 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ∈ 𝒫 (ℝ
↑𝑚 𝑋)) |
106 | | elpwi 4201 |
. . . . . . . . 9
⊢ ((𝐶‘𝑛) ∈ 𝒫 (ℝ
↑𝑚 𝑋) → (𝐶‘𝑛) ⊆ (ℝ ↑𝑚
𝑋)) |
107 | 105, 106 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ⊆ (ℝ ↑𝑚
𝑋)) |
108 | 103, 104,
107 | syl2anc 694 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → (𝐶‘𝑛) ⊆ (ℝ ↑𝑚
𝑋)) |
109 | 102, 108 | ovncl 41102 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶‘𝑛)) ∈ (0[,]+∞)) |
110 | 57 | fveq2d 6233 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘∅)) |
111 | 1 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
𝑋 ∈
Fin) |
112 | 111 | ovn0 41101 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
((voln*‘𝑋)‘∅) = 0) |
113 | 110, 112 | eqtrd 2685 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
((voln*‘𝑋)‘(𝐶‘𝑛)) = 0) |
114 | 98, 100, 101, 109, 113 | sge0ss 40947 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) =
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛))))) |
115 | 114 | eqcomd 2657 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) =
(Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛))))) |
116 | 80, 5 | eqsstrd 3672 |
. . . . . 6
⊢ (𝜑 → (𝐶‘1) ⊆ (ℝ
↑𝑚 𝑋)) |
117 | 1, 116 | ovncl 41102 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈
(0[,]+∞)) |
118 | 92, 17 | eqsstrd 3672 |
. . . . . 6
⊢ (𝜑 → (𝐶‘2) ⊆ (ℝ
↑𝑚 𝑋)) |
119 | 1, 118 | ovncl 41102 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈
(0[,]+∞)) |
120 | 73 | fveq2d 6233 |
. . . . 5
⊢ (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘1))) |
121 | 74 | fveq2d 6233 |
. . . . 5
⊢ (𝑛 = 2 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘2))) |
122 | 82 | a1i 11 |
. . . . 5
⊢ (𝜑 → 1 ≠ 2) |
123 | 79, 91, 117, 119, 120, 121, 122 | sge0pr 40929 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒
((voln*‘𝑋)‘(𝐶‘2)))) |
124 | 80 | fveq2d 6233 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴)) |
125 | 92 | fveq2d 6233 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵)) |
126 | 124, 125 | oveq12d 6708 |
. . . 4
⊢ (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒
((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))) |
127 | 115, 123,
126 | 3eqtrd 2689 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))) |
128 | 97, 127 | breq12d 4698 |
. 2
⊢ (𝜑 → (((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) ↔ ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))) |
129 | 36, 128 | mpbid 222 |
1
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))) |