| Step | Hyp | Ref
| Expression |
| 1 | | retop 24782 |
. . . 4
⊢
(topGen‘ran (,)) ∈ Top |
| 2 | | 0cld 23046 |
. . . 4
⊢
((topGen‘ran (,)) ∈ Top → ∅ ∈
(Clsd‘(topGen‘ran (,)))) |
| 3 | 1, 2 | ax-mp 5 |
. . 3
⊢ ∅
∈ (Clsd‘(topGen‘ran (,))) |
| 4 | | simpl3 1194 |
. . . . 5
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘𝐴)) |
| 5 | | fveq2 6906 |
. . . . . 6
⊢ (𝐴 = ∅ →
(vol*‘𝐴) =
(vol*‘∅)) |
| 6 | 5 | adantl 481 |
. . . . 5
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) →
(vol*‘𝐴) =
(vol*‘∅)) |
| 7 | 4, 6 | breqtrd 5169 |
. . . 4
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 <
(vol*‘∅)) |
| 8 | | 0ss 4400 |
. . . 4
⊢ ∅
⊆ 𝐴 |
| 9 | 7, 8 | jctil 519 |
. . 3
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) → (∅
⊆ 𝐴 ∧ 𝑀 <
(vol*‘∅))) |
| 10 | | sseq1 4009 |
. . . . 5
⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴)) |
| 11 | | fveq2 6906 |
. . . . . 6
⊢ (𝑠 = ∅ →
(vol*‘𝑠) =
(vol*‘∅)) |
| 12 | 11 | breq2d 5155 |
. . . . 5
⊢ (𝑠 = ∅ → (𝑀 < (vol*‘𝑠) ↔ 𝑀 <
(vol*‘∅))) |
| 13 | 10, 12 | anbi12d 632 |
. . . 4
⊢ (𝑠 = ∅ → ((𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)) ↔ (∅ ⊆ 𝐴 ∧ 𝑀 <
(vol*‘∅)))) |
| 14 | 13 | rspcev 3622 |
. . 3
⊢ ((∅
∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅))) →
∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 15 | 3, 9, 14 | sylancr 587 |
. 2
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) → ∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 16 | | mblfinlem1 37664 |
. . . 4
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 17 | 16 | 3ad2antl1 1186 |
. . 3
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 ≠ ∅) →
∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 18 | | simpl3 1194 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑀 < (vol*‘𝐴)) |
| 19 | | f1ofo 6855 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 20 | | rnco2 6273 |
. . . . . . . . . . . . . . . . 17
⊢ ran ([,]
∘ 𝑓) = ([,] “
ran 𝑓) |
| 21 | | forn 6823 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran 𝑓 = {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 22 | 21 | imaeq2d 6078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] “ ran 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 23 | 20, 22 | eqtrid 2789 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 24 | 23 | unieqd 4920 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran
([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 25 | 19, 24 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran
([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 26 | 25 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran
([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 27 | | oveq1 7438 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦))) |
| 28 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1)) |
| 29 | 28 | oveq1d 7446 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦))) |
| 30 | 27, 29 | opeq12d 4881 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑢 → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 = 〈(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))〉) |
| 31 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣)) |
| 32 | 31 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣))) |
| 33 | 31 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣))) |
| 34 | 32, 33 | opeq12d 4881 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑣 → 〈(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))〉 = 〈(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))〉) |
| 35 | 30, 34 | cbvmpov 7528 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) = (𝑢 ∈ ℤ, 𝑣 ∈ ℕ0 ↦
〈(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))〉) |
| 36 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧)) |
| 37 | 36 | sseq1d 4015 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐))) |
| 38 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑧 → (𝑎 = 𝑐 ↔ 𝑧 = 𝑐)) |
| 39 | 37, 38 | imbi12d 344 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐))) |
| 40 | 39 | ralbidv 3178 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑧 → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐))) |
| 41 | 40 | cbvrabv 3447 |
. . . . . . . . . . . . . . 15
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} = {𝑧 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)} |
| 42 | | ssrab2 4080 |
. . . . . . . . . . . . . . . 16
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
| 43 | 42 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
{𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
| 44 | 35, 41, 43 | dyadmbllem 25634 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = ∪ ([,]
“ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 45 | 44 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ([,]
“ {𝑏 ∈ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴}) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
| 46 | 26, 45 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran
([,] ∘ 𝑓) = ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴})) |
| 47 | | opnmbllem0 37663 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ (topGen‘ran (,))
→ ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴) |
| 48 | 47 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴) |
| 49 | 48 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ([,]
“ {𝑏 ∈ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴}) = 𝐴) |
| 50 | 46, 49 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran
([,] ∘ 𝑓) = 𝐴) |
| 51 | 50 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘∪ ran ([,] ∘ 𝑓)) = (vol*‘𝐴)) |
| 52 | | f1of 6848 |
. . . . . . . . . . . . 13
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 53 | | ssrab2 4080 |
. . . . . . . . . . . . . 14
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} |
| 54 | 35 | dyadf 25626 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶( ≤ ∩ (ℝ ×
ℝ)) |
| 55 | | frn 6743 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ ( ≤ ∩
(ℝ × ℝ))) |
| 56 | 54, 55 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ ( ≤ ∩
(ℝ × ℝ)) |
| 57 | 42, 56 | sstri 3993 |
. . . . . . . . . . . . . 14
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ( ≤ ∩ (ℝ ×
ℝ)) |
| 58 | 53, 57 | sstri 3993 |
. . . . . . . . . . . . 13
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ( ≤ ∩ (ℝ ×
ℝ)) |
| 59 | | fss 6752 |
. . . . . . . . . . . . 13
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ( ≤ ∩ (ℝ ×
ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
| 60 | 52, 58, 59 | sylancl 586 |
. . . . . . . . . . . 12
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
| 61 | 53, 42 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . 20
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
| 62 | | ffvelcdm 7101 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 63 | 61, 62 | sselid 3981 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
| 64 | 63 | adantrr 717 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
| 65 | | ffvelcdm 7101 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 66 | 61, 65 | sselid 3981 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
| 67 | 66 | adantrl 716 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
| 68 | 35 | dyaddisj 25631 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∧ (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
| 69 | 64, 67, 68 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
| 70 | 52, 69 | sylan 580 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
| 71 | | df-3or 1088 |
. . . . . . . . . . . . . . . 16
⊢
((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
| 72 | 70, 71 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
| 73 | | elrabi 3687 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) |
| 74 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑎 = (𝑓‘𝑚) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑚))) |
| 75 | 74 | sseq1d 4015 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑎 = (𝑓‘𝑚) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐))) |
| 76 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑎 = (𝑓‘𝑚) → (𝑎 = 𝑐 ↔ (𝑓‘𝑚) = 𝑐)) |
| 77 | 75, 76 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑎 = (𝑓‘𝑚) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐))) |
| 78 | 77 | ralbidv 3178 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑎 = (𝑓‘𝑚) → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐))) |
| 79 | 78 | elrab 3692 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ ((𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐))) |
| 80 | 79 | simprbi 496 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)) |
| 81 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = (𝑓‘𝑧) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑧))) |
| 82 | 81 | sseq2d 4016 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = (𝑓‘𝑧) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)))) |
| 83 | | eqeq2 2749 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = (𝑓‘𝑧) → ((𝑓‘𝑚) = 𝑐 ↔ (𝑓‘𝑚) = (𝑓‘𝑧))) |
| 84 | 82, 83 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑐 = (𝑓‘𝑧) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧)))) |
| 85 | 84 | rspcva 3620 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
| 86 | 73, 80, 85 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
| 87 | | elrabi 3687 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) |
| 88 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑎 = (𝑓‘𝑧) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑧))) |
| 89 | 88 | sseq1d 4015 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑎 = (𝑓‘𝑧) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐))) |
| 90 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑎 = (𝑓‘𝑧) → (𝑎 = 𝑐 ↔ (𝑓‘𝑧) = 𝑐)) |
| 91 | 89, 90 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑎 = (𝑓‘𝑧) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐))) |
| 92 | 91 | ralbidv 3178 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑎 = (𝑓‘𝑧) → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐))) |
| 93 | 92 | elrab 3692 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐))) |
| 94 | 93 | simprbi 496 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)) |
| 95 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑐 = (𝑓‘𝑚) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑚))) |
| 96 | 95 | sseq2d 4016 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = (𝑓‘𝑚) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)))) |
| 97 | | eqeq2 2749 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = (𝑓‘𝑚) → ((𝑓‘𝑧) = 𝑐 ↔ (𝑓‘𝑧) = (𝑓‘𝑚))) |
| 98 | 96, 97 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = (𝑓‘𝑚) → ((([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚)))) |
| 99 | 98 | rspcva 3620 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚))) |
| 100 | 87, 94, 99 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚))) |
| 101 | | eqcom 2744 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓‘𝑧) = (𝑓‘𝑚) ↔ (𝑓‘𝑚) = (𝑓‘𝑧)) |
| 102 | 100, 101 | imbitrdi 251 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
| 103 | 86, 102 | jaod 860 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
| 104 | 62, 65, 103 | syl2an 596 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) ∧ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
| 105 | 104 | anandis 678 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
| 106 | 52, 105 | sylan 580 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
| 107 | | f1of1 6847 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ–1-1→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 108 | | f1veqaeq 7277 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ–1-1→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑓‘𝑚) = (𝑓‘𝑧) → 𝑚 = 𝑧)) |
| 109 | 107, 108 | sylan 580 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑓‘𝑚) = (𝑓‘𝑧) → 𝑚 = 𝑧)) |
| 110 | 106, 109 | syld 47 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → 𝑚 = 𝑧)) |
| 111 | 110 | orim1d 968 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) → (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))) |
| 112 | 72, 111 | mpd 15 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
| 113 | 112 | ralrimivva 3202 |
. . . . . . . . . . . . 13
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
| 114 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑧 → (𝑚 = 𝑝 ↔ 𝑧 = 𝑝)) |
| 115 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 = 𝑧 → ((,)‘(𝑓‘𝑚)) = ((,)‘(𝑓‘𝑧))) |
| 116 | 115 | ineq1d 4219 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑧 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝)))) |
| 117 | 116 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑧 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅ ↔ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
| 118 | 114, 117 | orbi12d 919 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑧 → ((𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))) |
| 119 | 118 | ralbidv 3178 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑧 → (∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))) |
| 120 | 119 | cbvralvw 3237 |
. . . . . . . . . . . . . 14
⊢
(∀𝑚 ∈
ℕ ∀𝑝 ∈
ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
| 121 | | eqeq2 2749 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑝 → (𝑚 = 𝑧 ↔ 𝑚 = 𝑝)) |
| 122 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 𝑝 → ((,)‘(𝑓‘𝑧)) = ((,)‘(𝑓‘𝑝))) |
| 123 | 122 | ineq2d 4220 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑝 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝)))) |
| 124 | 123 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑝 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
| 125 | 121, 124 | orbi12d 919 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑝 → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))) |
| 126 | 125 | cbvralvw 3237 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑧 ∈
ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
| 127 | 126 | ralbii 3093 |
. . . . . . . . . . . . . 14
⊢
(∀𝑚 ∈
ℕ ∀𝑧 ∈
ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
| 128 | 122 | disjor 5125 |
. . . . . . . . . . . . . 14
⊢
(Disj 𝑧
∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
| 129 | 120, 127,
128 | 3bitr4ri 304 |
. . . . . . . . . . . . 13
⊢
(Disj 𝑧
∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
| 130 | 113, 129 | sylibr 234 |
. . . . . . . . . . . 12
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧))) |
| 131 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − )
∘ 𝑓)) |
| 132 | 60, 130, 131 | uniiccvol 25615 |
. . . . . . . . . . 11
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (vol*‘∪ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ −
) ∘ 𝑓)),
ℝ*, < )) |
| 133 | 132 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘∪ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ −
) ∘ 𝑓)),
ℝ*, < )) |
| 134 | 51, 133 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘𝐴) = sup(ran seq1( + , ((abs ∘ −
) ∘ 𝑓)),
ℝ*, < )) |
| 135 | 18, 134 | breqtrd 5169 |
. . . . . . . 8
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑀 < sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < )) |
| 136 | | absf 15376 |
. . . . . . . . . . . 12
⊢
abs:ℂ⟶ℝ |
| 137 | | subf 11510 |
. . . . . . . . . . . 12
⊢ −
:(ℂ × ℂ)⟶ℂ |
| 138 | | fco 6760 |
. . . . . . . . . . . 12
⊢
((abs:ℂ⟶ℝ ∧ − :(ℂ ×
ℂ)⟶ℂ) → (abs ∘ − ):(ℂ ×
ℂ)⟶ℝ) |
| 139 | 136, 137,
138 | mp2an 692 |
. . . . . . . . . . 11
⊢ (abs
∘ − ):(ℂ × ℂ)⟶ℝ |
| 140 | | zre 12617 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℝ) |
| 141 | | 2re 12340 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℝ |
| 142 | | reexpcl 14119 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((2
∈ ℝ ∧ 𝑦
∈ ℕ0) → (2↑𝑦) ∈ ℝ) |
| 143 | 141, 142 | mpan 690 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ∈
ℝ) |
| 144 | | 2cn 12341 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℂ |
| 145 | | 2ne0 12370 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ≠
0 |
| 146 | | nn0z 12638 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℤ) |
| 147 | | expne0i 14135 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((2
∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0) |
| 148 | 144, 145,
146, 147 | mp3an12i 1467 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ≠
0) |
| 149 | 143, 148 | jca 511 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ℕ0
→ ((2↑𝑦) ∈
ℝ ∧ (2↑𝑦)
≠ 0)) |
| 150 | | redivcl 11986 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ (𝑥 / (2↑𝑦)) ∈
ℝ) |
| 151 | | peano2re 11434 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
| 152 | | redivcl 11986 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 + 1) ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℝ) |
| 153 | 151, 152 | syl3an1 1164 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℝ) |
| 154 | 150, 153 | opelxpd 5724 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
| 155 | 154 | 3expb 1121 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ℝ ∧
((2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0))
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
| 156 | 140, 149,
155 | syl2an 596 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
| 157 | 156 | rgen2 3199 |
. . . . . . . . . . . . . . . . 17
⊢
∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ) |
| 158 | | eqid 2737 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
| 159 | 158 | fmpo 8093 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ × ℝ)
↔ (𝑥 ∈ ℤ,
𝑦 ∈
ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ)) |
| 160 | 157, 159 | mpbi 230 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ) |
| 161 | | frn 6743 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℝ
× ℝ)) |
| 162 | 160, 161 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℝ
× ℝ) |
| 163 | 42, 162 | sstri 3993 |
. . . . . . . . . . . . . 14
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ ×
ℝ) |
| 164 | 53, 163 | sstri 3993 |
. . . . . . . . . . . . 13
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ ×
ℝ) |
| 165 | | ax-resscn 11212 |
. . . . . . . . . . . . . 14
⊢ ℝ
⊆ ℂ |
| 166 | | xpss12 5700 |
. . . . . . . . . . . . . 14
⊢ ((ℝ
⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ)
⊆ (ℂ × ℂ)) |
| 167 | 165, 165,
166 | mp2an 692 |
. . . . . . . . . . . . 13
⊢ (ℝ
× ℝ) ⊆ (ℂ × ℂ) |
| 168 | 164, 167 | sstri 3993 |
. . . . . . . . . . . 12
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℂ ×
ℂ) |
| 169 | | fss 6752 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℂ × ℂ)) →
𝑓:ℕ⟶(ℂ
× ℂ)) |
| 170 | 168, 169 | mpan2 691 |
. . . . . . . . . . 11
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶(ℂ ×
ℂ)) |
| 171 | | fco 6760 |
. . . . . . . . . . 11
⊢ (((abs
∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ ×
ℂ)) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ) |
| 172 | 139, 170,
171 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((abs ∘ − ) ∘
𝑓):ℕ⟶ℝ) |
| 173 | | nnuz 12921 |
. . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) |
| 174 | | 1z 12647 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℤ |
| 175 | 174 | a1i 11 |
. . . . . . . . . . 11
⊢ (((abs
∘ − ) ∘ 𝑓):ℕ⟶ℝ → 1 ∈
ℤ) |
| 176 | | ffvelcdm 7101 |
. . . . . . . . . . 11
⊢ ((((abs
∘ − ) ∘ 𝑓):ℕ⟶ℝ ∧ 𝑛 ∈ ℕ) → (((abs
∘ − ) ∘ 𝑓)‘𝑛) ∈ ℝ) |
| 177 | 173, 175,
176 | serfre 14072 |
. . . . . . . . . 10
⊢ (((abs
∘ − ) ∘ 𝑓):ℕ⟶ℝ → seq1( + ,
((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ) |
| 178 | | frn 6743 |
. . . . . . . . . . 11
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( +
, ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ) |
| 179 | | ressxr 11305 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℝ* |
| 180 | 178, 179 | sstrdi 3996 |
. . . . . . . . . 10
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( +
, ((abs ∘ − ) ∘ 𝑓)) ⊆
ℝ*) |
| 181 | 52, 172, 177, 180 | 4syl 19 |
. . . . . . . . 9
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran seq1( + , ((abs ∘ −
) ∘ 𝑓)) ⊆
ℝ*) |
| 182 | | rexr 11307 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℝ → 𝑀 ∈
ℝ*) |
| 183 | 182 | 3ad2ant2 1135 |
. . . . . . . . 9
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
𝑀 ∈
ℝ*) |
| 184 | | supxrlub 13367 |
. . . . . . . . 9
⊢ ((ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* ∧ 𝑀 ∈ ℝ*)
→ (𝑀 < sup(ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔
∃𝑧 ∈ ran seq1( +
, ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧)) |
| 185 | 181, 183,
184 | syl2anr 597 |
. . . . . . . 8
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (𝑀 < sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ) ↔ ∃𝑧 ∈ ran seq1( + , ((abs ∘ − )
∘ 𝑓))𝑀 < 𝑧)) |
| 186 | 135, 185 | mpbid 232 |
. . . . . . 7
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑧 ∈ ran seq1( + , ((abs ∘ − )
∘ 𝑓))𝑀 < 𝑧) |
| 187 | | seqfn 14054 |
. . . . . . . . . 10
⊢ (1 ∈
ℤ → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn
(ℤ≥‘1)) |
| 188 | 174, 187 | ax-mp 5 |
. . . . . . . . 9
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) Fn
(ℤ≥‘1) |
| 189 | 173 | fneq2i 6666 |
. . . . . . . . 9
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ↔ seq1( + , ((abs ∘
− ) ∘ 𝑓)) Fn
(ℤ≥‘1)) |
| 190 | 188, 189 | mpbir 231 |
. . . . . . . 8
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) Fn ℕ |
| 191 | | breq2 5147 |
. . . . . . . . 9
⊢ (𝑧 = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛) → (𝑀 < 𝑧 ↔ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛))) |
| 192 | 191 | rexrn 7107 |
. . . . . . . 8
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)) Fn ℕ → (∃𝑧 ∈ ran seq1( + , ((abs
∘ − ) ∘ 𝑓))𝑀 < 𝑧 ↔ ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛))) |
| 193 | 190, 192 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑧 ∈ ran
seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧 ↔ ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛)) |
| 194 | 186, 193 | sylib 218 |
. . . . . 6
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛)) |
| 195 | 60 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
| 196 | | 0le0 12367 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ≤
0 |
| 197 | | df-br 5144 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ≤ 0
↔ 〈0, 0〉 ∈ ≤ ) |
| 198 | 196, 197 | mpbi 230 |
. . . . . . . . . . . . . . . . 17
⊢ 〈0,
0〉 ∈ ≤ |
| 199 | | 0re 11263 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
ℝ |
| 200 | | opelxpi 5722 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℝ ∧ 0 ∈ ℝ) → 〈0, 0〉 ∈ (ℝ
× ℝ)) |
| 201 | 199, 199,
200 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢ 〈0,
0〉 ∈ (ℝ × ℝ) |
| 202 | | elin 3967 |
. . . . . . . . . . . . . . . . 17
⊢ (〈0,
0〉 ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (〈0, 0〉
∈ ≤ ∧ 〈0, 0〉 ∈ (ℝ ×
ℝ))) |
| 203 | 198, 201,
202 | mpbir2an 711 |
. . . . . . . . . . . . . . . 16
⊢ 〈0,
0〉 ∈ ( ≤ ∩ (ℝ × ℝ)) |
| 204 | | ifcl 4571 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓‘𝑧) ∈ ( ≤ ∩ (ℝ ×
ℝ)) ∧ 〈0, 0〉 ∈ ( ≤ ∩ (ℝ ×
ℝ))) → if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈ ( ≤ ∩
(ℝ × ℝ))) |
| 205 | 195, 203,
204 | sylancl 586 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈ ( ≤ ∩
(ℝ × ℝ))) |
| 206 | 205 | fmpttd 7135 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)):ℕ⟶( ≤
∩ (ℝ × ℝ))) |
| 207 | | df-ov 7434 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0(,)0) =
((,)‘〈0, 0〉) |
| 208 | | iooid 13415 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0(,)0) =
∅ |
| 209 | 207, 208 | eqtr3i 2767 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((,)‘〈0, 0〉) = ∅ |
| 210 | 209 | ineq1i 4216 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘〈0, 0〉) ∩ ((,)‘(𝑓‘𝑧))) = (∅ ∩ ((,)‘(𝑓‘𝑧))) |
| 211 | | 0in 4397 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (∅
∩ ((,)‘(𝑓‘𝑧))) = ∅ |
| 212 | 210, 211 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘〈0, 0〉) ∩ ((,)‘(𝑓‘𝑧))) = ∅ |
| 213 | 212 | olci 867 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑧 ∨ (((,)‘〈0, 0〉) ∩
((,)‘(𝑓‘𝑧))) = ∅) |
| 214 | | ineq1 4213 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
(((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧)))) |
| 215 | 214 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅)) |
| 216 | 215 | orbi2d 916 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) =
∅))) |
| 217 | | ineq1 4213 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((,)‘〈0, 0〉) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
(((,)‘〈0, 0〉) ∩ ((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧)))) |
| 218 | 217 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘〈0, 0〉) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
((((,)‘〈0, 0〉) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅)) |
| 219 | 218 | orbi2d 916 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘〈0, 0〉) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
((𝑚 = 𝑧 ∨ (((,)‘〈0, 0〉) ∩
((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) =
∅))) |
| 220 | 216, 219 | ifboth 4565 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ∧ (𝑚 = 𝑧 ∨ (((,)‘〈0, 0〉) ∩
((,)‘(𝑓‘𝑧))) = ∅)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅)) |
| 221 | 112, 213,
220 | sylancl 586 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅)) |
| 222 | 209 | ineq2i 4217 |
. . . . . . . . . . . . . . . . . . 19
⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
∅) |
| 223 | | in0 4395 |
. . . . . . . . . . . . . . . . . . 19
⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
∅) = ∅ |
| 224 | 222, 223 | eqtri 2765 |
. . . . . . . . . . . . . . . . . 18
⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅ |
| 225 | 224 | olci 867 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅) |
| 226 | | ineq2 4214 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
(if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0,
0〉)))) |
| 227 | 226 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
((if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
| 228 | 227 | orbi2d 916 |
. . . . . . . . . . . . . . . . . 18
⊢
(((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅))) |
| 229 | | ineq2 4214 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘〈0, 0〉) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
(if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0,
0〉)))) |
| 230 | 229 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘〈0, 0〉) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
((if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
| 231 | 230 | orbi2d 916 |
. . . . . . . . . . . . . . . . . 18
⊢
(((,)‘〈0, 0〉) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅))) |
| 232 | 228, 231 | ifboth 4565 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅) ∧ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
| 233 | 221, 225,
232 | sylancl 586 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
| 234 | 233 | ralrimivva 3202 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
| 235 | | disjeq2 5114 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑚 ∈
ℕ ((,)‘((𝑧
∈ ℕ ↦ if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
(Disj 𝑚 ∈
ℕ ((,)‘((𝑧
∈ ℕ ↦ if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) ↔ Disj 𝑚 ∈ ℕ if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0,
0〉)))) |
| 236 | | eleq1w 2824 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑚 → (𝑧 ∈ (1...𝑛) ↔ 𝑚 ∈ (1...𝑛))) |
| 237 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑚 → (𝑓‘𝑧) = (𝑓‘𝑚)) |
| 238 | 236, 237 | ifbieq1d 4550 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑚 → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉)) |
| 239 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) |
| 240 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓‘𝑚) ∈ V |
| 241 | | opex 5469 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 〈0,
0〉 ∈ V |
| 242 | 240, 241 | ifex 4576 |
. . . . . . . . . . . . . . . . . . . 20
⊢ if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉) ∈
V |
| 243 | 238, 239,
242 | fvmpt 7016 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈ ℕ → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉)) |
| 244 | 243 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ℕ →
((,)‘((𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉))) |
| 245 | | fvif 6922 |
. . . . . . . . . . . . . . . . . 18
⊢
((,)‘if(𝑚
∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0,
0〉)) |
| 246 | 244, 245 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ →
((,)‘((𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0,
0〉))) |
| 247 | 235, 246 | mprg 3067 |
. . . . . . . . . . . . . . . 16
⊢
(Disj 𝑚
∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) ↔ Disj 𝑚 ∈ ℕ if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0,
0〉))) |
| 248 | | eleq1w 2824 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑧 → (𝑚 ∈ (1...𝑛) ↔ 𝑧 ∈ (1...𝑛))) |
| 249 | 248, 115 | ifbieq1d 4550 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑧 → if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0,
0〉))) |
| 250 | 249 | disjor 5125 |
. . . . . . . . . . . . . . . 16
⊢
(Disj 𝑚
∈ ℕ if(𝑚 ∈
(1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ↔
∀𝑚 ∈ ℕ
∀𝑧 ∈ ℕ
(𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
| 251 | 247, 250 | bitri 275 |
. . . . . . . . . . . . . . 15
⊢
(Disj 𝑚
∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
| 252 | 234, 251 | sylibr 234 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → Disj 𝑚 ∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
| 253 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) = seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
| 254 | 206, 252,
253 | uniiccvol 25615 |
. . . . . . . . . . . . 13
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) = sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))), ℝ*,
< )) |
| 255 | 254 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) = sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))), ℝ*,
< )) |
| 256 | | rexpssxrxp 11306 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
| 257 | 164, 256 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . 20
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ* ×
ℝ*) |
| 258 | 257, 65 | sselid 3981 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℝ* ×
ℝ*)) |
| 259 | | 0xr 11308 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ∈
ℝ* |
| 260 | | opelxpi 5722 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((0
∈ ℝ* ∧ 0 ∈ ℝ*) → 〈0,
0〉 ∈ (ℝ* ×
ℝ*)) |
| 261 | 259, 259,
260 | mp2an 692 |
. . . . . . . . . . . . . . . . . . 19
⊢ 〈0,
0〉 ∈ (ℝ* ×
ℝ*) |
| 262 | | ifcl 4571 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓‘𝑧) ∈ (ℝ* ×
ℝ*) ∧ 〈0, 0〉 ∈ (ℝ* ×
ℝ*)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈
(ℝ* × ℝ*)) |
| 263 | 258, 261,
262 | sylancl 586 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈
(ℝ* × ℝ*)) |
| 264 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
| 265 | | iccf 13488 |
. . . . . . . . . . . . . . . . . . . 20
⊢
[,]:(ℝ* × ℝ*)⟶𝒫
ℝ* |
| 266 | 265 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → [,]:(ℝ* ×
ℝ*)⟶𝒫 ℝ*) |
| 267 | 266 | feqmptd 6977 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → [,] = (𝑚 ∈ (ℝ* ×
ℝ*) ↦ ([,]‘𝑚))) |
| 268 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) → ([,]‘𝑚) = ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
| 269 | 263, 264,
267, 268 | fmptco 7149 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
| 270 | 52, 269 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
| 271 | 270 | rneqd 5949 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = ran (𝑧 ∈ ℕ ↦
([,]‘if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
| 272 | 271 | unieqd 4920 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran
([,] ∘ (𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
| 273 | | peano2nn 12278 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℕ) |
| 274 | 273, 173 | eleqtrdi 2851 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
(ℤ≥‘1)) |
| 275 | | fzouzsplit 13734 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 + 1) ∈
(ℤ≥‘1) → (ℤ≥‘1) =
((1..^(𝑛 + 1)) ∪
(ℤ≥‘(𝑛 + 1)))) |
| 276 | 274, 275 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ →
(ℤ≥‘1) = ((1..^(𝑛 + 1)) ∪
(ℤ≥‘(𝑛 + 1)))) |
| 277 | 173, 276 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → ℕ =
((1..^(𝑛 + 1)) ∪
(ℤ≥‘(𝑛 + 1)))) |
| 278 | | nnz 12634 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
| 279 | | fzval3 13773 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℤ →
(1...𝑛) = (1..^(𝑛 + 1))) |
| 280 | 278, 279 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ →
(1...𝑛) = (1..^(𝑛 + 1))) |
| 281 | 280 | uneq1d 4167 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ →
((1...𝑛) ∪
(ℤ≥‘(𝑛 + 1))) = ((1..^(𝑛 + 1)) ∪
(ℤ≥‘(𝑛 + 1)))) |
| 282 | 277, 281 | eqtr4d 2780 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → ℕ =
((1...𝑛) ∪
(ℤ≥‘(𝑛 + 1)))) |
| 283 | | fvif 6922 |
. . . . . . . . . . . . . . . . . 18
⊢
([,]‘if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉)) |
| 284 | 283 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ →
([,]‘if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉))) |
| 285 | 282, 284 | iuneq12d 5021 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = ∪ 𝑧 ∈ ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1)))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉))) |
| 286 | | fvex 6919 |
. . . . . . . . . . . . . . . . 17
⊢
([,]‘if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) ∈
V |
| 287 | 286 | dfiun3 5980 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
| 288 | | iunxun 5094 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑧 ∈ ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1)))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) = (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) ∪
∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉))) |
| 289 | 285, 287,
288 | 3eqtr3g 2800 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) ∪
∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉)))) |
| 290 | | iftrue 4531 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) =
([,]‘(𝑓‘𝑧))) |
| 291 | 290 | iuneq2i 5013 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) |
| 292 | 291 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))) |
| 293 | | uznfz 13650 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈
(ℤ≥‘(𝑛 + 1)) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1))) |
| 294 | 293 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1))) |
| 295 | | nncn 12274 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
| 296 | | ax-1cn 11213 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 1 ∈
ℂ |
| 297 | | pncan 11514 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑛 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑛 + 1)
− 1) = 𝑛) |
| 298 | 295, 296,
297 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛) |
| 299 | 298 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ ℕ →
(1...((𝑛 + 1) − 1)) =
(1...𝑛)) |
| 300 | 299 | eleq2d 2827 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ℕ → (𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ 𝑧 ∈ (1...𝑛))) |
| 301 | 300 | notbid 318 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → (¬
𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬
𝑧 ∈ (1...𝑛))) |
| 302 | 301 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘(𝑛 + 1))) → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛))) |
| 303 | 294, 302 | mpbid 232 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...𝑛)) |
| 304 | 303 | iffalsed 4536 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘(𝑛 + 1))) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) =
([,]‘〈0, 0〉)) |
| 305 | 304 | iuneq2dv 5016 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) = ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)) |
| 306 | 292, 305 | uneq12d 4169 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) ∪
∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉))) =
(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉))) |
| 307 | 289, 306 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉))) |
| 308 | 272, 307 | sylan9eq 2797 |
. . . . . . . . . . . . 13
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉))) |
| 309 | 308 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) =
(vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)))) |
| 310 | | xrltso 13183 |
. . . . . . . . . . . . . . 15
⊢ < Or
ℝ* |
| 311 | 310 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → < Or
ℝ*) |
| 312 | | elnnuz 12922 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈
(ℤ≥‘1)) |
| 313 | 312 | biimpi 216 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
(ℤ≥‘1)) |
| 314 | 313 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
(ℤ≥‘1)) |
| 315 | | elfznn 13593 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 ∈ (1...𝑛) → 𝑢 ∈ ℕ) |
| 316 | 172 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑢 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑢) ∈ ℝ) |
| 317 | 315, 316 | sylan2 593 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑢 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑢) ∈ ℝ) |
| 318 | 317 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑢 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑢) ∈ ℝ) |
| 319 | | readdcl 11238 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ) |
| 320 | 319 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ (𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑢 + 𝑣) ∈ ℝ) |
| 321 | 314, 318,
320 | seqcl 14063 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ) |
| 322 | 321 | rexrd 11311 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) ∈
ℝ*) |
| 323 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ (1...𝑛) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
| 324 | | iftrue 4531 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈ (1...𝑛) → if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉) = (𝑓‘𝑚)) |
| 325 | 238, 324 | sylan9eqr 2799 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑚 ∈ (1...𝑛) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = (𝑓‘𝑚)) |
| 326 | | elfznn 13593 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ (1...𝑛) → 𝑚 ∈ ℕ) |
| 327 | 240 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ (1...𝑛) → (𝑓‘𝑚) ∈ V) |
| 328 | 323, 325,
326, 327 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 ∈ (1...𝑛) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = (𝑓‘𝑚)) |
| 329 | 328 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = (𝑓‘𝑚)) |
| 330 | 329 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = ((abs ∘ −
)‘(𝑓‘𝑚))) |
| 331 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓‘𝑧) ∈ V |
| 332 | 331, 241 | ifex 4576 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈
V |
| 333 | 332, 239 | fnmpti 6711 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) Fn
ℕ |
| 334 | | fvco2 7006 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
| 335 | 333, 326,
334 | sylancr 587 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈ (1...𝑛) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
| 336 | 335 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
| 337 | | ffn 6736 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓 Fn ℕ) |
| 338 | | fvco2 7006 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs
∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚))) |
| 339 | 337, 326,
338 | syl2an 596 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚))) |
| 340 | 330, 336,
339 | 3eqtr4d 2787 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = (((abs ∘ − )
∘ 𝑓)‘𝑚)) |
| 341 | 340 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = (((abs ∘ − )
∘ 𝑓)‘𝑚)) |
| 342 | 314, 341 | seqfveq 14067 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 343 | 174 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 1 ∈ ℤ) |
| 344 | 168, 65 | sselid 3981 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℂ ×
ℂ)) |
| 345 | | 0cn 11253 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 ∈
ℂ |
| 346 | | opelxpi 5722 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((0
∈ ℂ ∧ 0 ∈ ℂ) → 〈0, 0〉 ∈ (ℂ
× ℂ)) |
| 347 | 345, 345,
346 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 〈0,
0〉 ∈ (ℂ × ℂ) |
| 348 | | ifcl 4571 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓‘𝑧) ∈ (ℂ × ℂ) ∧
〈0, 0〉 ∈ (ℂ × ℂ)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈ (ℂ ×
ℂ)) |
| 349 | 344, 347,
348 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈ (ℂ ×
ℂ)) |
| 350 | 349 | fmpttd 7135 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)):ℕ⟶(ℂ
× ℂ)) |
| 351 | | fco 6760 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((abs
∘ − ):(ℂ × ℂ)⟶ℝ ∧ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)):ℕ⟶(ℂ
× ℂ)) → ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0,
0〉))):ℕ⟶ℝ) |
| 352 | 139, 350,
351 | sylancr 587 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0,
0〉))):ℕ⟶ℝ) |
| 353 | 352 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ (𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) ∈
ℝ) |
| 354 | 173, 343,
353 | serfre 14072 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0,
0〉)))):ℕ⟶ℝ) |
| 355 | 354 | ffnd 6737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) Fn
ℕ) |
| 356 | | fnfvelrn 7100 |
. . . . . . . . . . . . . . . 16
⊢ ((seq1( +
, ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) Fn ℕ ∧
𝑛 ∈ ℕ) →
(seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) |
| 357 | 355, 356 | sylan 580 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) |
| 358 | 342, 357 | eqeltrrd 2842 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) ∈ ran seq1( + , ((abs ∘ −
) ∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) |
| 359 | 354 | frnd 6744 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran seq1( + , ((abs ∘ −
) ∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) ⊆
ℝ) |
| 360 | 359 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) ⊆
ℝ) |
| 361 | 360 | sselda 3983 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) → 𝑚 ∈
ℝ) |
| 362 | 321 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) → (seq1( + ,
((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ) |
| 363 | | readdcl 11238 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑚 + 𝑢) ∈ ℝ) |
| 364 | 363 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ (𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ)) → (𝑚 + 𝑢) ∈ ℝ) |
| 365 | | recn 11245 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 ∈ ℝ → 𝑚 ∈
ℂ) |
| 366 | | recn 11245 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑢 ∈ ℝ → 𝑢 ∈
ℂ) |
| 367 | | recn 11245 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑣 ∈ ℝ → 𝑣 ∈
ℂ) |
| 368 | | addass 11242 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣))) |
| 369 | 365, 366,
367, 368 | syl3an 1161 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣))) |
| 370 | 369 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ (𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣))) |
| 371 | | nnltp1le 12674 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 < 𝑡 ↔ (𝑛 + 1) ≤ 𝑡)) |
| 372 | 371 | biimpa 476 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑛 + 1) ≤ 𝑡) |
| 373 | 273 | nnzd 12640 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℤ) |
| 374 | | nnz 12634 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 ∈ ℕ → 𝑡 ∈
ℤ) |
| 375 | | eluz 12892 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑛 + 1) ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ∈
(ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡)) |
| 376 | 373, 374,
375 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑡 ∈
(ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡)) |
| 377 | 376 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡)) |
| 378 | 372, 377 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1))) |
| 379 | 378 | adantlll 718 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1))) |
| 380 | 313 | ad3antlr 731 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑛 ∈
(ℤ≥‘1)) |
| 381 | | simplll 775 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 382 | | elfznn 13593 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℕ) |
| 383 | 381, 382,
353 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) ∈
ℝ) |
| 384 | 364, 370,
379, 380, 383 | seqsplit 14076 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = ((seq1( + , ((abs ∘
− ) ∘ (𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡))) |
| 385 | 342 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 386 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℤ) |
| 387 | 386 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ) |
| 388 | | 0red 11264 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ) |
| 389 | 273 | nnred 12281 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℝ) |
| 390 | 389 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ) |
| 391 | 386 | zred 12722 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℝ) |
| 392 | 391 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ) |
| 393 | 273 | nngt0d 12315 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑛 ∈ ℕ → 0 <
(𝑛 + 1)) |
| 394 | 393 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1)) |
| 395 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → (𝑛 + 1) ≤ 𝑚) |
| 396 | 395 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚) |
| 397 | 388, 390,
392, 394, 396 | ltletrd 11421 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚) |
| 398 | | elnnz 12623 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ ℕ ↔ (𝑚 ∈ ℤ ∧ 0 <
𝑚)) |
| 399 | 387, 397,
398 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ) |
| 400 | 333, 399,
334 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
| 401 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
| 402 | | nnre 12273 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
| 403 | 402 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 ∈ ℝ) |
| 404 | 389 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ) |
| 405 | 391 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ) |
| 406 | 402 | ltp1d 12198 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1)) |
| 407 | 406 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < (𝑛 + 1)) |
| 408 | 395 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚) |
| 409 | 403, 404,
405, 407, 408 | ltletrd 11421 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < 𝑚) |
| 410 | 409 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → 𝑛 < 𝑚) |
| 411 | 403, 405 | ltnled 11408 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛)) |
| 412 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑚 = 𝑧 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛)) |
| 413 | 412 | equcoms 2019 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑧 = 𝑚 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛)) |
| 414 | 413 | notbid 318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑧 = 𝑚 → (¬ 𝑚 ≤ 𝑛 ↔ ¬ 𝑧 ≤ 𝑛)) |
| 415 | 411, 414 | sylan9bb 509 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → (𝑛 < 𝑚 ↔ ¬ 𝑧 ≤ 𝑛)) |
| 416 | 410, 415 | mpbid 232 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ≤ 𝑛) |
| 417 | | elfzle2 13568 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ≤ 𝑛) |
| 418 | 416, 417 | nsyl 140 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ∈ (1...𝑛)) |
| 419 | 418 | iffalsed 4536 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = 〈0,
0〉) |
| 420 | 386 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ) |
| 421 | | 0red 11264 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ) |
| 422 | 393 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1)) |
| 423 | 421, 404,
405, 422, 408 | ltletrd 11421 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚) |
| 424 | 420, 423,
398 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ) |
| 425 | 241 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 〈0, 0〉 ∈
V) |
| 426 | 401, 419,
424, 425 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = 〈0,
0〉) |
| 427 | 426 | ad4ant14 752 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = 〈0,
0〉) |
| 428 | 427 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = ((abs ∘ −
)‘〈0, 0〉)) |
| 429 | 400, 428 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘〈0, 0〉)) |
| 430 | | fvco3 7008 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((
− :(ℂ × ℂ)⟶ℂ ∧ 〈0, 0〉 ∈
(ℂ × ℂ)) → ((abs ∘ − )‘〈0,
0〉) = (abs‘( − ‘〈0, 0〉))) |
| 431 | 137, 347,
430 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((abs
∘ − )‘〈0, 0〉) = (abs‘( −
‘〈0, 0〉)) |
| 432 | | df-ov 7434 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (0
− 0) = ( − ‘〈0, 0〉) |
| 433 | | 0m0e0 12386 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (0
− 0) = 0 |
| 434 | 432, 433 | eqtr3i 2767 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ( −
‘〈0, 0〉) = 0 |
| 435 | 434 | fveq2i 6909 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(abs‘( − ‘〈0, 0〉)) =
(abs‘0) |
| 436 | | abs0 15324 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(abs‘0) = 0 |
| 437 | 435, 436 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(abs‘( − ‘〈0, 0〉)) = 0 |
| 438 | 431, 437 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((abs
∘ − )‘〈0, 0〉) = 0 |
| 439 | 429, 438 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = 0) |
| 440 | | elfzuz 13560 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) |
| 441 | | c0ex 11255 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ 0 ∈
V |
| 442 | 441 | fvconst2 7224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑚 ∈
(ℤ≥‘(𝑛 + 1)) →
(((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0) |
| 443 | 440, 442 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) →
(((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0) |
| 444 | 443 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) →
(((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0) |
| 445 | 439, 444 | eqtr4d 2780 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) =
(((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚)) |
| 446 | 378, 445 | seqfveq 14067 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = (seq(𝑛 + 1)( + ,
((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡)) |
| 447 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1)) |
| 448 | 447 | ser0 14095 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 ∈
(ℤ≥‘(𝑛 + 1)) → (seq(𝑛 + 1)( + ,
((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡) = 0) |
| 449 | 378, 448 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + ,
((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡) = 0) |
| 450 | 446, 449 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = 0) |
| 451 | 450 | adantlll 718 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = 0) |
| 452 | 385, 451 | oveq12d 7449 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡)) = ((seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛) + 0)) |
| 453 | 172 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑚) ∈ ℝ) |
| 454 | 326, 453 | sylan2 593 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) ∈ ℝ) |
| 455 | 454 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) ∈ ℝ) |
| 456 | | readdcl 11238 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑚 + 𝑣) ∈ ℝ) |
| 457 | 456 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑚 + 𝑣) ∈ ℝ) |
| 458 | 314, 455,
457 | seqcl 14063 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ) |
| 459 | 458 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ∈
ℝ) |
| 460 | 459 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ∈
ℂ) |
| 461 | 460 | addridd 11461 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) + 0) = (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛)) |
| 462 | 452, 461 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡)) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 463 | 384, 462 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 464 | 453 | ad5ant15 759 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑚) ∈ ℝ) |
| 465 | 326, 464 | sylan2 593 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) ∈ ℝ) |
| 466 | 380, 465,
364 | seqcl 14063 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ∈
ℝ) |
| 467 | 466 | leidd 11829 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 468 | 463, 467 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 469 | | elnnuz 12922 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑡 ∈ ℕ ↔ 𝑡 ∈
(ℤ≥‘1)) |
| 470 | 469 | biimpi 216 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 ∈ ℕ → 𝑡 ∈
(ℤ≥‘1)) |
| 471 | 470 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑡 ∈
(ℤ≥‘1)) |
| 472 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
| 473 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 = 𝑚) |
| 474 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ (1...𝑡) → 1 ≤ 𝑚) |
| 475 | 474 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 1 ≤ 𝑚) |
| 476 | 382 | nnred 12281 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℝ) |
| 477 | 476 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℝ) |
| 478 | | nnre 12273 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑡 ∈ ℕ → 𝑡 ∈
ℝ) |
| 479 | 478 | ad3antlr 731 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ∈ ℝ) |
| 480 | 402 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑛 ∈ ℝ) |
| 481 | | elfzle2 13568 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ≤ 𝑡) |
| 482 | 481 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑡) |
| 483 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ≤ 𝑛) |
| 484 | 477, 479,
480, 482, 483 | letrd 11418 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑛) |
| 485 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℤ) |
| 486 | 278 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ ℤ) |
| 487 | | elfz 13553 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑚 ∈ ℤ ∧ 1 ∈
ℤ ∧ 𝑛 ∈
ℤ) → (𝑚 ∈
(1...𝑛) ↔ (1 ≤
𝑚 ∧ 𝑚 ≤ 𝑛))) |
| 488 | 174, 487 | mp3an2 1451 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛))) |
| 489 | 485, 486,
488 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛))) |
| 490 | 475, 484,
489 | mpbir2and 713 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛)) |
| 491 | 490 | ad5ant2345 1372 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛)) |
| 492 | 491 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑚 ∈ (1...𝑛)) |
| 493 | 473, 492 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 ∈ (1...𝑛)) |
| 494 | | iftrue 4531 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = (𝑓‘𝑧)) |
| 495 | 493, 494 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = (𝑓‘𝑧)) |
| 496 | 237 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → (𝑓‘𝑧) = (𝑓‘𝑚)) |
| 497 | 495, 496 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = (𝑓‘𝑚)) |
| 498 | 382 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℕ) |
| 499 | 240 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑓‘𝑚) ∈ V) |
| 500 | 472, 497,
498, 499 | fvmptd 7023 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = (𝑓‘𝑚)) |
| 501 | 500 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = ((abs ∘ −
)‘(𝑓‘𝑚))) |
| 502 | 333, 382,
334 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈ (1...𝑡) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
| 503 | 502 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
| 504 | | simplll 775 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
| 505 | | fvco3 7008 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚))) |
| 506 | 504, 382,
505 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚))) |
| 507 | 501, 503,
506 | 3eqtr4d 2787 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = (((abs ∘ − )
∘ 𝑓)‘𝑚)) |
| 508 | 471, 507 | seqfveq 14067 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑡)) |
| 509 | | eluz 12892 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈
(ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛)) |
| 510 | 374, 278,
509 | syl2anr 597 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 ∈
(ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛)) |
| 511 | 510 | biimpar 477 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ (ℤ≥‘𝑡)) |
| 512 | 511 | adantlll 718 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ (ℤ≥‘𝑡)) |
| 513 | 504, 326,
453 | syl2an 596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) ∈ ℝ) |
| 514 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℤ) |
| 515 | 514 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℤ) |
| 516 | | 0red 11264 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ∈ ℝ) |
| 517 | | peano2nn 12278 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈
ℕ) |
| 518 | 517 | nnred 12281 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈
ℝ) |
| 519 | 518 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ∈ ℝ) |
| 520 | 514 | zred 12722 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℝ) |
| 521 | 520 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℝ) |
| 522 | 517 | nngt0d 12315 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 ∈ ℕ → 0 <
(𝑡 + 1)) |
| 523 | 522 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < (𝑡 + 1)) |
| 524 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → (𝑡 + 1) ≤ 𝑚) |
| 525 | 524 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ≤ 𝑚) |
| 526 | 516, 519,
521, 523, 525 | ltletrd 11421 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < 𝑚) |
| 527 | 515, 526,
398 | sylanbrc 583 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ) |
| 528 | 527 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑡 ∈ ℕ ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ) |
| 529 | 528 | adantlll 718 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ) |
| 530 | 170 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ (ℂ ×
ℂ)) |
| 531 | | ffvelcdm 7101 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((
− :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) → (
− ‘(𝑓‘𝑚)) ∈ ℂ) |
| 532 | 137, 530,
531 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → ( −
‘(𝑓‘𝑚)) ∈
ℂ) |
| 533 | 532 | absge0d 15483 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → 0 ≤ (abs‘(
− ‘(𝑓‘𝑚)))) |
| 534 | | fvco3 7008 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((
− :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) →
((abs ∘ − )‘(𝑓‘𝑚)) = (abs‘( − ‘(𝑓‘𝑚)))) |
| 535 | 137, 530,
534 | sylancr 587 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → ((abs ∘ −
)‘(𝑓‘𝑚)) = (abs‘( −
‘(𝑓‘𝑚)))) |
| 536 | 505, 535 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑚) = (abs‘( − ‘(𝑓‘𝑚)))) |
| 537 | 533, 536 | breqtrrd 5171 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝑓)‘𝑚)) |
| 538 | 537 | ad5ant15 759 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝑓)‘𝑚)) |
| 539 | 529, 538 | syldan 591 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ≤ (((abs ∘ − )
∘ 𝑓)‘𝑚)) |
| 540 | 471, 512,
513, 539 | sermono 14075 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 541 | 508, 540 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 542 | 402 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑛 ∈ ℝ) |
| 543 | 478 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℝ) |
| 544 | 468, 541,
542, 543 | ltlecasei 11369 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 545 | 544 | ralrimiva 3146 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∀𝑡 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 546 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = (seq1( + , ((abs ∘
− ) ∘ (𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) → (𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛))) |
| 547 | 546 | ralrn 7108 |
. . . . . . . . . . . . . . . . . . 19
⊢ (seq1( +
, ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) Fn ℕ →
(∀𝑚 ∈ ran seq1(
+ , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛))) |
| 548 | 355, 547 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (∀𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛))) |
| 549 | 548 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (∀𝑚 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛))) |
| 550 | 545, 549 | mpbird 257 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∀𝑚 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛)) |
| 551 | 550 | r19.21bi 3251 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) → 𝑚 ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
| 552 | 361, 362,
551 | lensymd 11412 |
. . . . . . . . . . . . . 14
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) → ¬ (seq1(
+ , ((abs ∘ − ) ∘ 𝑓))‘𝑛) < 𝑚) |
| 553 | 311, 322,
358, 552 | supmax 9507 |
. . . . . . . . . . . . 13
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))), ℝ*,
< ) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)) |
| 554 | 52, 553 | sylan 580 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))), ℝ*,
< ) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)) |
| 555 | 255, 309,
554 | 3eqtr3rd 2786 |
. . . . . . . . . . 11
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) = (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)))) |
| 556 | | elfznn 13593 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ∈ ℕ) |
| 557 | 164, 65 | sselid 3981 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℝ ×
ℝ)) |
| 558 | | 1st2nd2 8053 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
(𝑓‘𝑧) = 〈(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))〉) |
| 559 | 558 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
([,]‘(𝑓‘𝑧)) =
([,]‘〈(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))〉)) |
| 560 | | df-ov 7434 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) = ([,]‘〈(1st
‘(𝑓‘𝑧)), (2nd
‘(𝑓‘𝑧))〉) |
| 561 | 559, 560 | eqtr4di 2795 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
([,]‘(𝑓‘𝑧)) = ((1st
‘(𝑓‘𝑧))[,](2nd
‘(𝑓‘𝑧)))) |
| 562 | | xp1st 8046 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
(1st ‘(𝑓‘𝑧)) ∈ ℝ) |
| 563 | | xp2nd 8047 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
(2nd ‘(𝑓‘𝑧)) ∈ ℝ) |
| 564 | | iccssre 13469 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd
‘(𝑓‘𝑧)) ∈ ℝ) →
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ) |
| 565 | 562, 563,
564 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ) |
| 566 | 561, 565 | eqsstrd 4018 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
([,]‘(𝑓‘𝑧)) ⊆
ℝ) |
| 567 | 557, 566 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
| 568 | 52, 556, 567 | syl2an 596 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
| 569 | 568 | ralrimiva 3146 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
| 570 | | iunss 5045 |
. . . . . . . . . . . . . 14
⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
| 571 | 569, 570 | sylibr 234 |
. . . . . . . . . . . . 13
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪
𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
| 572 | 571 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
| 573 | | uzid 12893 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 + 1) ∈ ℤ →
(𝑛 + 1) ∈
(ℤ≥‘(𝑛 + 1))) |
| 574 | | ne0i 4341 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 + 1) ∈
(ℤ≥‘(𝑛 + 1)) →
(ℤ≥‘(𝑛 + 1)) ≠ ∅) |
| 575 | | iunconst 5001 |
. . . . . . . . . . . . . . . 16
⊢
((ℤ≥‘(𝑛 + 1)) ≠ ∅ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) = ([,]‘〈0, 0〉)) |
| 576 | 373, 573,
574, 575 | 4syl 19 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) = ([,]‘〈0, 0〉)) |
| 577 | | iccid 13432 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
ℝ* → (0[,]0) = {0}) |
| 578 | 259, 577 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ (0[,]0) =
{0} |
| 579 | | df-ov 7434 |
. . . . . . . . . . . . . . . 16
⊢ (0[,]0) =
([,]‘〈0, 0〉) |
| 580 | 578, 579 | eqtr3i 2767 |
. . . . . . . . . . . . . . 15
⊢ {0} =
([,]‘〈0, 0〉) |
| 581 | 576, 580 | eqtr4di 2795 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) = {0}) |
| 582 | | snssi 4808 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
ℝ → {0} ⊆ ℝ) |
| 583 | 199, 582 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ {0}
⊆ ℝ |
| 584 | 581, 583 | eqsstrdi 4028 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) ⊆ ℝ) |
| 585 | 584 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) ⊆ ℝ) |
| 586 | 581 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ →
(vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)) = (vol*‘{0})) |
| 587 | 586 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)) = (vol*‘{0})) |
| 588 | | ovolsn 25530 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
ℝ → (vol*‘{0}) = 0) |
| 589 | 199, 588 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(vol*‘{0}) = 0 |
| 590 | 587, 589 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)) = 0) |
| 591 | | ovolunnul 25535 |
. . . . . . . . . . . 12
⊢
((∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ ∧ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) ⊆ ℝ ∧ (vol*‘∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0, 0〉)) = 0)
→ (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0, 0〉))) =
(vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
| 592 | 572, 585,
590, 591 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0, 0〉))) =
(vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
| 593 | 555, 592 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
| 594 | 593 | breq2d 5155 |
. . . . . . . . 9
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
| 595 | 594 | biimpd 229 |
. . . . . . . 8
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) → 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
| 596 | 595 | reximdva 3168 |
. . . . . . 7
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
| 597 | 596 | adantl 481 |
. . . . . 6
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
| 598 | 194, 597 | mpd 15 |
. . . . 5
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
| 599 | | fzfi 14013 |
. . . . . . . . . 10
⊢
(1...𝑛) ∈
Fin |
| 600 | | icccld 24787 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd
‘(𝑓‘𝑧)) ∈ ℝ) →
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 601 | 562, 563,
600 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 602 | 561, 601 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
([,]‘(𝑓‘𝑧)) ∈
(Clsd‘(topGen‘ran (,)))) |
| 603 | 557, 602 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 604 | 556, 603 | sylan2 593 |
. . . . . . . . . . 11
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 605 | 604 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 606 | | uniretop 24783 |
. . . . . . . . . . 11
⊢ ℝ =
∪ (topGen‘ran (,)) |
| 607 | 606 | iuncld 23053 |
. . . . . . . . . 10
⊢
(((topGen‘ran (,)) ∈ Top ∧ (1...𝑛) ∈ Fin ∧ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 608 | 1, 599, 605, 607 | mp3an12i 1467 |
. . . . . . . . 9
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪
𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 609 | 608 | adantr 480 |
. . . . . . . 8
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
| 610 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = (𝑓‘𝑧) → ([,]‘𝑏) = ([,]‘(𝑓‘𝑧))) |
| 611 | 610 | sseq1d 4015 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑓‘𝑧) → (([,]‘𝑏) ⊆ 𝐴 ↔ ([,]‘(𝑓‘𝑧)) ⊆ 𝐴)) |
| 612 | 611 | elrab 3692 |
. . . . . . . . . . . . . 14
⊢ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ↔ ((𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∧ ([,]‘(𝑓‘𝑧)) ⊆ 𝐴)) |
| 613 | 612 | simprbi 496 |
. . . . . . . . . . . . 13
⊢ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
| 614 | 65, 73, 613 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
| 615 | 556, 614 | sylan2 593 |
. . . . . . . . . . 11
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
| 616 | 615 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
| 617 | | iunss 5045 |
. . . . . . . . . 10
⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
| 618 | 616, 617 | sylibr 234 |
. . . . . . . . 9
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪
𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
| 619 | 618 | adantr 480 |
. . . . . . . 8
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
| 620 | | simprr 773 |
. . . . . . . 8
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
| 621 | | sseq1 4009 |
. . . . . . . . . 10
⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑠 ⊆ 𝐴 ↔ ∪
𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴)) |
| 622 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (vol*‘𝑠) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
| 623 | 622 | breq2d 5155 |
. . . . . . . . . 10
⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
| 624 | 621, 623 | anbi12d 632 |
. . . . . . . . 9
⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → ((𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)) ↔ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))) |
| 625 | 624 | rspcev 3622 |
. . . . . . . 8
⊢
((∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,)))
∧ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 626 | 609, 619,
620, 625 | syl12anc 837 |
. . . . . . 7
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 627 | 52, 626 | sylan 580 |
. . . . . 6
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 628 | 627 | adantll 714 |
. . . . 5
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 629 | 598, 628 | rexlimddv 3161 |
. . . 4
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 630 | 629 | adantlr 715 |
. . 3
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 ≠ ∅) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 631 | 17, 630 | exlimddv 1935 |
. 2
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 ≠ ∅) →
∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
| 632 | 15, 631 | pm2.61dane 3029 |
1
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |