Proof of Theorem ovnsubadd2lem
| Step | Hyp | Ref
| Expression |
| 1 | | ovnsubadd2lem.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ Fin) |
| 2 | | iftrue 4531 |
. . . . . . . 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 7466 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ
↑m 𝑋)
∈ V) |
| 5 | | ovnsubadd2lem.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) |
| 6 | 4, 5 | ssexd 5324 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ V) |
| 7 | 6, 5 | elpwd 4606 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 8 | 7 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 = 1) → 𝐴 ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 9 | 3, 8 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 10 | 9 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 11 | | simpl 482 |
. . . . . . . . . . 11
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → ¬ 𝑛 = 1) |
| 12 | 11 | iffalsed 4536 |
. . . . . . . . . 10
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅)) |
| 13 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → 𝑛 = 2) |
| 14 | 13 | iftrued 4533 |
. . . . . . . . . 10
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = 𝐵) |
| 15 | 12, 14 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((¬
𝑛 = 1 ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵) |
| 16 | 15 | adantll 714 |
. . . . . . . 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 5324 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ V) |
| 19 | 18, 17 | elpwd 4606 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 20 | 19 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → 𝐵 ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 21 | 16, 20 | eqeltrd 2841 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 22 | 21 | adantllr 719 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) ∧ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 23 | | simpl 482 |
. . . . . . . . . 10
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 1) |
| 24 | 23 | iffalsed 4536 |
. . . . . . . . 9
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅)) |
| 25 | | simpr 484 |
. . . . . . . . . 10
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → ¬ 𝑛 = 2) |
| 26 | 25 | iffalsed 4536 |
. . . . . . . . 9
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 2, 𝐵, ∅) = ∅) |
| 27 | 24, 26 | eqtrd 2777 |
. . . . . . . 8
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅) |
| 28 | | 0elpw 5356 |
. . . . . . . . 9
⊢ ∅
∈ 𝒫 (ℝ ↑m 𝑋) |
| 29 | 28 | a1i 11 |
. . . . . . . 8
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → ∅ ∈
𝒫 (ℝ ↑m 𝑋)) |
| 30 | 27, 29 | eqeltrd 2841 |
. . . . . . 7
⊢ ((¬
𝑛 = 1 ∧ ¬ 𝑛 = 2) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 31 | 30 | adantll 714 |
. . . . . 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 813 |
. . . . 5
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ ¬ 𝑛 = 1) → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 33 | 10, 32 | pm2.61dan 813 |
. . . 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 7134 |
. . 3
⊢ (𝜑 → 𝐶:ℕ⟶𝒫 (ℝ
↑m 𝑋)) |
| 36 | 1, 35 | ovnsubadd 46587 |
. 2
⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) ≤
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛))))) |
| 37 | | eldifi 4131 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ 𝑛 ∈
ℕ) |
| 38 | 37 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
𝑛 ∈
ℕ) |
| 39 | | eldifn 4132 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ ¬ 𝑛 ∈ {1,
2}) |
| 40 | | vex 3484 |
. . . . . . . . . . . . . . . . 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 4654 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑛 ∈ {1, 2} →
𝑛 ≠ 1) |
| 44 | 43 | neneqd 2945 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑛 ∈ {1, 2} →
¬ 𝑛 =
1) |
| 45 | 39, 44 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ ¬ 𝑛 =
1) |
| 46 | 41, 42 | nelpr2 4653 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑛 ∈ {1, 2} →
𝑛 ≠ 2) |
| 47 | 46 | neneqd 2945 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑛 ∈ {1, 2} →
¬ 𝑛 =
2) |
| 48 | 39, 47 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ ¬ 𝑛 =
2) |
| 49 | 45, 48, 27 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = ∅) |
| 50 | | 0ex 5307 |
. . . . . . . . . . . . 13
⊢ ∅
∈ V |
| 51 | 50 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (ℕ ∖ {1, 2})
→ ∅ ∈ V) |
| 52 | 49, 51 | eqeltrd 2841 |
. . . . . . . . . . 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 7027 |
. . . . . . . . . 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 584 |
. . . . . . . . 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 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
(𝐶‘𝑛) = ∅) |
| 58 | 57 | ralrimiva 3146 |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈ (ℕ ∖ {1, 2})(𝐶‘𝑛) = ∅) |
| 59 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑛(ℕ ∖ {1, 2}) |
| 60 | 59 | iunxdif3 5095 |
. . . . . . 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 2743 |
. . . . 5
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = ∪ 𝑛 ∈ (ℕ ∖
(ℕ ∖ {1, 2}))(𝐶‘𝑛)) |
| 63 | | 1nn 12277 |
. . . . . . . . . 10
⊢ 1 ∈
ℕ |
| 64 | | 2nn 12339 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ |
| 65 | 63, 64 | pm3.2i 470 |
. . . . . . . . 9
⊢ (1 ∈
ℕ ∧ 2 ∈ ℕ) |
| 66 | | prssi 4821 |
. . . . . . . . 9
⊢ ((1
∈ ℕ ∧ 2 ∈ ℕ) → {1, 2} ⊆
ℕ) |
| 67 | 65, 66 | ax-mp 5 |
. . . . . . . 8
⊢ {1, 2}
⊆ ℕ |
| 68 | | dfss4 4269 |
. . . . . . . 8
⊢ ({1, 2}
⊆ ℕ ↔ (ℕ ∖ (ℕ ∖ {1, 2})) = {1,
2}) |
| 69 | 67, 68 | mpbi 230 |
. . . . . . 7
⊢ (ℕ
∖ (ℕ ∖ {1, 2})) = {1, 2} |
| 70 | | iuneq1 5008 |
. . . . . . 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 6906 |
. . . . . . . . 9
⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1)) |
| 74 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑛 = 2 → (𝐶‘𝑛) = (𝐶‘2)) |
| 75 | 73, 74 | iunxprg 5096 |
. . . . . . . 8
⊢ ((1
∈ ℕ ∧ 2 ∈ ℕ) → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = ((𝐶‘1) ∪ (𝐶‘2))) |
| 76 | 63, 64, 75 | mp2an 692 |
. . . . . . 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 | 63 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℕ) |
| 79 | 34, 2, 78, 6 | fvmptd3 7039 |
. . . . . . 7
⊢ (𝜑 → (𝐶‘1) = 𝐴) |
| 80 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑛 = 2 → 𝑛 = 2) |
| 81 | | 1ne2 12474 |
. . . . . . . . . . . . . 14
⊢ 1 ≠
2 |
| 82 | 81 | necomi 2995 |
. . . . . . . . . . . . 13
⊢ 2 ≠
1 |
| 83 | 82 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑛 = 2 → 2 ≠
1) |
| 84 | 80, 83 | eqnetrd 3008 |
. . . . . . . . . . 11
⊢ (𝑛 = 2 → 𝑛 ≠ 1) |
| 85 | 84 | neneqd 2945 |
. . . . . . . . . 10
⊢ (𝑛 = 2 → ¬ 𝑛 = 1) |
| 86 | 85 | iffalsed 4536 |
. . . . . . . . 9
⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = if(𝑛 = 2, 𝐵, ∅)) |
| 87 | | iftrue 4531 |
. . . . . . . . 9
⊢ (𝑛 = 2 → if(𝑛 = 2, 𝐵, ∅) = 𝐵) |
| 88 | 86, 87 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝑛 = 2 → if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)) = 𝐵) |
| 89 | 64 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 2 ∈
ℕ) |
| 90 | 34, 88, 89, 18 | fvmptd3 7039 |
. . . . . . 7
⊢ (𝜑 → (𝐶‘2) = 𝐵) |
| 91 | 79, 90 | uneq12d 4169 |
. . . . . 6
⊢ (𝜑 → ((𝐶‘1) ∪ (𝐶‘2)) = (𝐴 ∪ 𝐵)) |
| 92 | | eqidd 2738 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)) |
| 93 | 77, 91, 92 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → ∪ 𝑛 ∈ {1, 2} (𝐶‘𝑛) = (𝐴 ∪ 𝐵)) |
| 94 | 62, 72, 93 | 3eqtrd 2781 |
. . . 4
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐶‘𝑛) = (𝐴 ∪ 𝐵)) |
| 95 | 94 | fveq2d 6910 |
. . 3
⊢ (𝜑 → ((voln*‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐴 ∪ 𝐵))) |
| 96 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑛𝜑 |
| 97 | | nnex 12272 |
. . . . . . 7
⊢ ℕ
∈ V |
| 98 | 97 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℕ ∈
V) |
| 99 | 67 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {1, 2} ⊆
ℕ) |
| 100 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑋 ∈ Fin) |
| 101 | | simpl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝜑) |
| 102 | 99 | sselda 3983 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → 𝑛 ∈ ℕ) |
| 103 | 35 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 104 | | elpwi 4607 |
. . . . . . . . 9
⊢ ((𝐶‘𝑛) ∈ 𝒫 (ℝ
↑m 𝑋)
→ (𝐶‘𝑛) ⊆ (ℝ
↑m 𝑋)) |
| 105 | 103, 104 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋)) |
| 106 | 101, 102,
105 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → (𝐶‘𝑛) ⊆ (ℝ ↑m 𝑋)) |
| 107 | 100, 106 | ovncl 46582 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ {1, 2}) → ((voln*‘𝑋)‘(𝐶‘𝑛)) ∈ (0[,]+∞)) |
| 108 | 57 | fveq2d 6910 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘∅)) |
| 109 | 1 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
𝑋 ∈
Fin) |
| 110 | 109 | ovn0 46581 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
((voln*‘𝑋)‘∅) = 0) |
| 111 | 108, 110 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ ∖ {1, 2})) →
((voln*‘𝑋)‘(𝐶‘𝑛)) = 0) |
| 112 | 96, 98, 99, 107, 111 | sge0ss 46427 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) =
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛))))) |
| 113 | 112 | eqcomd 2743 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) =
(Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛))))) |
| 114 | 79, 5 | eqsstrd 4018 |
. . . . . 6
⊢ (𝜑 → (𝐶‘1) ⊆ (ℝ
↑m 𝑋)) |
| 115 | 1, 114 | ovncl 46582 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) ∈
(0[,]+∞)) |
| 116 | 90, 17 | eqsstrd 4018 |
. . . . . 6
⊢ (𝜑 → (𝐶‘2) ⊆ (ℝ
↑m 𝑋)) |
| 117 | 1, 116 | ovncl 46582 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) ∈
(0[,]+∞)) |
| 118 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑛 = 1 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘1))) |
| 119 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑛 = 2 → ((voln*‘𝑋)‘(𝐶‘𝑛)) = ((voln*‘𝑋)‘(𝐶‘2))) |
| 120 | 81 | a1i 11 |
. . . . 5
⊢ (𝜑 → 1 ≠ 2) |
| 121 | 78, 89, 115, 117, 118, 119, 120 | sge0pr 46409 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ {1, 2} ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘(𝐶‘1)) +𝑒
((voln*‘𝑋)‘(𝐶‘2)))) |
| 122 | 79 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘1)) = ((voln*‘𝑋)‘𝐴)) |
| 123 | 90 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐶‘2)) = ((voln*‘𝑋)‘𝐵)) |
| 124 | 122, 123 | oveq12d 7449 |
. . . 4
⊢ (𝜑 → (((voln*‘𝑋)‘(𝐶‘1)) +𝑒
((voln*‘𝑋)‘(𝐶‘2))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))) |
| 125 | 113, 121,
124 | 3eqtrd 2781 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐶‘𝑛)))) = (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))) |
| 126 | 95, 125 | breq12d 5156 |
. 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)))) |
| 127 | 36, 126 | mpbid 232 |
1
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))) |