Step |
Hyp |
Ref |
Expression |
1 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
2 |
|
0cld |
⊢ ( ( topGen ‘ ran (,) ) ∈ Top → ∅ ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ) |
3 |
1 2
|
ax-mp |
⊢ ∅ ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) |
4 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → 𝑀 < ( vol* ‘ 𝐴 ) ) |
5 |
|
fveq2 |
⊢ ( 𝐴 = ∅ → ( vol* ‘ 𝐴 ) = ( vol* ‘ ∅ ) ) |
6 |
5
|
adantl |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → ( vol* ‘ 𝐴 ) = ( vol* ‘ ∅ ) ) |
7 |
4 6
|
breqtrd |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → 𝑀 < ( vol* ‘ ∅ ) ) |
8 |
|
0ss |
⊢ ∅ ⊆ 𝐴 |
9 |
7 8
|
jctil |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → ( ∅ ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ ∅ ) ) ) |
10 |
|
sseq1 |
⊢ ( 𝑠 = ∅ → ( 𝑠 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴 ) ) |
11 |
|
fveq2 |
⊢ ( 𝑠 = ∅ → ( vol* ‘ 𝑠 ) = ( vol* ‘ ∅ ) ) |
12 |
11
|
breq2d |
⊢ ( 𝑠 = ∅ → ( 𝑀 < ( vol* ‘ 𝑠 ) ↔ 𝑀 < ( vol* ‘ ∅ ) ) ) |
13 |
10 12
|
anbi12d |
⊢ ( 𝑠 = ∅ → ( ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) ↔ ( ∅ ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ ∅ ) ) ) ) |
14 |
13
|
rspcev |
⊢ ( ( ∅ ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ∧ ( ∅ ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ ∅ ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) ) |
15 |
3 9 14
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) ) |
16 |
|
mblfinlem1 |
⊢ ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( 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 |
⊢ ran ( [,] ∘ 𝑓 ) = ( [,] “ ran 𝑓 ) |
21 |
|
forn |
⊢ ( 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 |
⊢ ( 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
|
syl5eq |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( ( ( 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 |
⊢ ( 𝑥 = 𝑢 → ( 𝑥 / ( 2 ↑ 𝑦 ) ) = ( 𝑢 / ( 2 ↑ 𝑦 ) ) ) |
28 |
|
oveq1 |
⊢ ( 𝑥 = 𝑢 → ( 𝑥 + 1 ) = ( 𝑢 + 1 ) ) |
29 |
28
|
oveq1d |
⊢ ( 𝑥 = 𝑢 → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) = ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑦 ) ) ) |
30 |
27 29
|
opeq12d |
⊢ ( 𝑥 = 𝑢 → 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 = 〈 ( 𝑢 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) |
31 |
|
oveq2 |
⊢ ( 𝑦 = 𝑣 → ( 2 ↑ 𝑦 ) = ( 2 ↑ 𝑣 ) ) |
32 |
31
|
oveq2d |
⊢ ( 𝑦 = 𝑣 → ( 𝑢 / ( 2 ↑ 𝑦 ) ) = ( 𝑢 / ( 2 ↑ 𝑣 ) ) ) |
33 |
31
|
oveq2d |
⊢ ( 𝑦 = 𝑣 → ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑦 ) ) = ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑣 ) ) ) |
34 |
32 33
|
opeq12d |
⊢ ( 𝑦 = 𝑣 → 〈 ( 𝑢 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 = 〈 ( 𝑢 / ( 2 ↑ 𝑣 ) ) , ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑣 ) ) 〉 ) |
35 |
30 34
|
cbvmpov |
⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) = ( 𝑢 ∈ ℤ , 𝑣 ∈ ℕ0 ↦ 〈 ( 𝑢 / ( 2 ↑ 𝑣 ) ) , ( ( 𝑢 + 1 ) / ( 2 ↑ 𝑣 ) ) 〉 ) |
36 |
|
fveq2 |
⊢ ( 𝑎 = 𝑧 → ( [,] ‘ 𝑎 ) = ( [,] ‘ 𝑧 ) ) |
37 |
36
|
sseq1d |
⊢ ( 𝑎 = 𝑧 → ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ 𝑧 ) ⊆ ( [,] ‘ 𝑐 ) ) ) |
38 |
|
eqeq1 |
⊢ ( 𝑎 = 𝑧 → ( 𝑎 = 𝑐 ↔ 𝑧 = 𝑐 ) ) |
39 |
37 38
|
imbi12d |
⊢ ( 𝑎 = 𝑧 → ( ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ( ( [,] ‘ 𝑧 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑧 = 𝑐 ) ) ) |
40 |
39
|
ralbidv |
⊢ ( 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 |
⊢ { 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 |
⊢ { 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( 𝐴 ∈ ( topGen ‘ ran (,) ) → ∪ ( [,] “ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ) = 𝐴 ) |
48 |
47
|
3ad2ant1 |
⊢ ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( 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 |
⊢ { 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 |
⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) : ( ℤ × ℕ0 ) ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) |
55 |
|
frn |
⊢ ( ( 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 |
⊢ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) ⊆ ( ≤ ∩ ( ℝ × ℝ ) ) |
57 |
42 56
|
sstri |
⊢ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ⊆ ( ≤ ∩ ( ℝ × ℝ ) ) |
58 |
53 57
|
sstri |
⊢ { 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ { 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 |
|
ffvelrn |
⊢ ( ( 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
|
sseldi |
⊢ ( ( 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 |
⊢ ( ( 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 |
|
ffvelrn |
⊢ ( ( 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
|
sseldi |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( [,] ‘ ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( [,] ‘ 𝑎 ) = ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) |
75 |
74
|
sseq1d |
⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) ) ) |
76 |
|
eqeq1 |
⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( 𝑎 = 𝑐 ↔ ( 𝑓 ‘ 𝑚 ) = 𝑐 ) ) |
77 |
75 76
|
imbi12d |
⊢ ( 𝑎 = ( 𝑓 ‘ 𝑚 ) → ( ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑚 ) = 𝑐 ) ) ) |
78 |
77
|
ralbidv |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑐 = ( 𝑓 ‘ 𝑧 ) → ( [,] ‘ 𝑐 ) = ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) |
82 |
81
|
sseq2d |
⊢ ( 𝑐 = ( 𝑓 ‘ 𝑧 ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) |
83 |
|
eqeq2 |
⊢ ( 𝑐 = ( 𝑓 ‘ 𝑧 ) → ( ( 𝑓 ‘ 𝑚 ) = 𝑐 ↔ ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) ) |
84 |
82 83
|
imbi12d |
⊢ ( 𝑐 = ( 𝑓 ‘ 𝑧 ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑚 ) = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) ) ) |
85 |
84
|
rspcva |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( [,] ‘ 𝑎 ) = ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) |
89 |
88
|
sseq1d |
⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) ) ) |
90 |
|
eqeq1 |
⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( 𝑎 = 𝑐 ↔ ( 𝑓 ‘ 𝑧 ) = 𝑐 ) ) |
91 |
89 90
|
imbi12d |
⊢ ( 𝑎 = ( 𝑓 ‘ 𝑧 ) → ( ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑧 ) = 𝑐 ) ) ) |
92 |
91
|
ralbidv |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑐 = ( 𝑓 ‘ 𝑚 ) → ( [,] ‘ 𝑐 ) = ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) |
96 |
95
|
sseq2d |
⊢ ( 𝑐 = ( 𝑓 ‘ 𝑚 ) → ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) ) ) |
97 |
|
eqeq2 |
⊢ ( 𝑐 = ( 𝑓 ‘ 𝑚 ) → ( ( 𝑓 ‘ 𝑧 ) = 𝑐 ↔ ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ) ) |
98 |
96 97
|
imbi12d |
⊢ ( 𝑐 = ( 𝑓 ‘ 𝑚 ) → ( ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓 ‘ 𝑧 ) = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓 ‘ 𝑚 ) ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ) ) ) |
99 |
98
|
rspcva |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ↔ ( 𝑓 ‘ 𝑚 ) = ( 𝑓 ‘ 𝑧 ) ) |
102 |
100 101
|
syl6ib |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( 𝑚 = 𝑧 → ( 𝑚 = 𝑝 ↔ 𝑧 = 𝑝 ) ) |
115 |
|
2fveq3 |
⊢ ( 𝑚 = 𝑧 → ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) = ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) |
116 |
115
|
ineq1d |
⊢ ( 𝑚 = 𝑧 → ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) ) |
117 |
116
|
eqeq1d |
⊢ ( 𝑚 = 𝑧 → ( ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ↔ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) |
118 |
114 117
|
orbi12d |
⊢ ( 𝑚 = 𝑧 → ( ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ↔ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) ) |
119 |
118
|
ralbidv |
⊢ ( 𝑚 = 𝑧 → ( ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ↔ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) ) |
120 |
119
|
cbvralvw |
⊢ ( ∀ 𝑚 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ↔ ∀ 𝑧 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) |
121 |
|
eqeq2 |
⊢ ( 𝑧 = 𝑝 → ( 𝑚 = 𝑧 ↔ 𝑚 = 𝑝 ) ) |
122 |
|
2fveq3 |
⊢ ( 𝑧 = 𝑝 → ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) = ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) |
123 |
122
|
ineq2d |
⊢ ( 𝑧 = 𝑝 → ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) ) |
124 |
123
|
eqeq1d |
⊢ ( 𝑧 = 𝑝 → ( ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ↔ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) |
125 |
121 124
|
orbi12d |
⊢ ( 𝑧 = 𝑝 → ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) ) |
126 |
125
|
cbvralvw |
⊢ ( ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) |
127 |
126
|
ralbii |
⊢ ( ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) |
128 |
122
|
disjor |
⊢ ( Disj 𝑧 ∈ ℕ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ↔ ∀ 𝑧 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑝 ) ) ) = ∅ ) ) |
129 |
120 127 128
|
3bitr4ri |
⊢ ( Disj 𝑧 ∈ ℕ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) ) |
130 |
113 129
|
sylibr |
⊢ ( 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 |
⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) |
132 |
60 130 131
|
uniiccvol |
⊢ ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) |
133 |
132
|
adantl |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) |
134 |
51 133
|
eqtr3d |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) |
135 |
18 134
|
breqtrd |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) |
136 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
137 |
|
subf |
⊢ − : ( ℂ × ℂ ) ⟶ ℂ |
138 |
|
fco |
⊢ ( ( abs : ℂ ⟶ ℝ ∧ − : ( ℂ × ℂ ) ⟶ ℂ ) → ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ) |
139 |
136 137 138
|
mp2an |
⊢ ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ |
140 |
|
zre |
⊢ ( 𝑥 ∈ ℤ → 𝑥 ∈ ℝ ) |
141 |
|
2re |
⊢ 2 ∈ ℝ |
142 |
|
reexpcl |
⊢ ( ( 2 ∈ ℝ ∧ 𝑦 ∈ ℕ0 ) → ( 2 ↑ 𝑦 ) ∈ ℝ ) |
143 |
141 142
|
mpan |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 ↑ 𝑦 ) ∈ ℝ ) |
144 |
|
2cn |
⊢ 2 ∈ ℂ |
145 |
|
2ne0 |
⊢ 2 ≠ 0 |
146 |
|
nn0z |
⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ ) |
147 |
|
expne0i |
⊢ ( ( 2 ∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ ) → ( 2 ↑ 𝑦 ) ≠ 0 ) |
148 |
144 145 146 147
|
mp3an12i |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 ↑ 𝑦 ) ≠ 0 ) |
149 |
143 148
|
jca |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) ) |
150 |
|
redivcl |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) → ( 𝑥 / ( 2 ↑ 𝑦 ) ) ∈ ℝ ) |
151 |
|
peano2re |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ ) |
152 |
|
redivcl |
⊢ ( ( ( 𝑥 + 1 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ∈ ℝ ) |
153 |
151 152
|
syl3an1 |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) → ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ∈ ℝ ) |
154 |
150 153
|
opelxpd |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) → 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ( ℝ × ℝ ) ) |
155 |
154
|
3expb |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( ( 2 ↑ 𝑦 ) ∈ ℝ ∧ ( 2 ↑ 𝑦 ) ≠ 0 ) ) → 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ( ℝ × ℝ ) ) |
156 |
140 149 155
|
syl2an |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0 ) → 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ( ℝ × ℝ ) ) |
157 |
156
|
rgen2 |
⊢ ∀ 𝑥 ∈ ℤ ∀ 𝑦 ∈ ℕ0 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ∈ ( ℝ × ℝ ) |
158 |
|
eqid |
⊢ ( 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 |
⊢ ( ∀ 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 |
⊢ ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) : ( ℤ × ℕ0 ) ⟶ ( ℝ × ℝ ) |
161 |
|
frn |
⊢ ( ( 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 |
⊢ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) ⊆ ( ℝ × ℝ ) |
163 |
42 162
|
sstri |
⊢ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ 〈 ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) 〉 ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ⊆ ( ℝ × ℝ ) |
164 |
53 163
|
sstri |
⊢ { 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 |
⊢ ℝ ⊆ ℂ |
166 |
|
xpss12 |
⊢ ( ( ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ ) → ( ℝ × ℝ ) ⊆ ( ℂ × ℂ ) ) |
167 |
165 165 166
|
mp2an |
⊢ ( ℝ × ℝ ) ⊆ ( ℂ × ℂ ) |
168 |
164 167
|
sstri |
⊢ { 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( ( ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ∧ 𝑓 : ℕ ⟶ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ ) |
172 |
139 170 171
|
sylancr |
⊢ ( 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 |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
174 |
|
1z |
⊢ 1 ∈ ℤ |
175 |
174
|
a1i |
⊢ ( ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ → 1 ∈ ℤ ) |
176 |
|
ffvelrn |
⊢ ( ( ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑛 ) ∈ ℝ ) |
177 |
173 175 176
|
serfre |
⊢ ( ( ( abs ∘ − ) ∘ 𝑓 ) : ℕ ⟶ ℝ → seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) : ℕ ⟶ ℝ ) |
178 |
|
frn |
⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) : ℕ ⟶ ℝ → ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ⊆ ℝ ) |
179 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
180 |
178 179
|
sstrdi |
⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) : ℕ ⟶ ℝ → ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ⊆ ℝ* ) |
181 |
52 172 177 180
|
4syl |
⊢ ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ⊆ ℝ* ) |
182 |
|
rexr |
⊢ ( 𝑀 ∈ ℝ → 𝑀 ∈ ℝ* ) |
183 |
182
|
3ad2ant2 |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → 𝑀 ∈ ℝ* ) |
184 |
|
supxrlub |
⊢ ( ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ⊆ ℝ* ∧ 𝑀 ∈ ℝ* ) → ( 𝑀 < sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ↔ ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ) ) |
185 |
181 183 184
|
syl2anr |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ↔ ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ) ) |
186 |
135 185
|
mpbid |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ) |
187 |
|
seqfn |
⊢ ( 1 ∈ ℤ → seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ( ℤ≥ ‘ 1 ) ) |
188 |
174 187
|
ax-mp |
⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ( ℤ≥ ‘ 1 ) |
189 |
173
|
fneq2i |
⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ℕ ↔ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ( ℤ≥ ‘ 1 ) ) |
190 |
188 189
|
mpbir |
⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ℕ |
191 |
|
breq2 |
⊢ ( 𝑧 = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) → ( 𝑀 < 𝑧 ↔ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) ) |
192 |
191
|
rexrn |
⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) Fn ℕ → ( ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ↔ ∃ 𝑛 ∈ ℕ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) ) |
193 |
190 192
|
ax-mp |
⊢ ( ∃ 𝑧 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) 𝑀 < 𝑧 ↔ ∃ 𝑛 ∈ ℕ 𝑀 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
194 |
186 193
|
sylib |
⊢ ( ( ( 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 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
195 |
60
|
ffvelrnda |
⊢ ( ( 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 |
⊢ 0 ≤ 0 |
197 |
|
df-br |
⊢ ( 0 ≤ 0 ↔ 〈 0 , 0 〉 ∈ ≤ ) |
198 |
196 197
|
mpbi |
⊢ 〈 0 , 0 〉 ∈ ≤ |
199 |
|
0re |
⊢ 0 ∈ ℝ |
200 |
|
opelxpi |
⊢ ( ( 0 ∈ ℝ ∧ 0 ∈ ℝ ) → 〈 0 , 0 〉 ∈ ( ℝ × ℝ ) ) |
201 |
199 199 200
|
mp2an |
⊢ 〈 0 , 0 〉 ∈ ( ℝ × ℝ ) |
202 |
|
elin |
⊢ ( 〈 0 , 0 〉 ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ↔ ( 〈 0 , 0 〉 ∈ ≤ ∧ 〈 0 , 0 〉 ∈ ( ℝ × ℝ ) ) ) |
203 |
198 201 202
|
mpbir2an |
⊢ 〈 0 , 0 〉 ∈ ( ≤ ∩ ( ℝ × ℝ ) ) |
204 |
|
ifcl |
⊢ ( ( ( 𝑓 ‘ 𝑧 ) ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 〈 0 , 0 〉 ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
205 |
195 203 204
|
sylancl |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( 0 (,) 0 ) = ( (,) ‘ 〈 0 , 0 〉 ) |
208 |
|
iooid |
⊢ ( 0 (,) 0 ) = ∅ |
209 |
207 208
|
eqtr3i |
⊢ ( (,) ‘ 〈 0 , 0 〉 ) = ∅ |
210 |
209
|
ineq1i |
⊢ ( ( (,) ‘ 〈 0 , 0 〉 ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( ∅ ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) |
211 |
|
0in |
⊢ ( ∅ ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ |
212 |
210 211
|
eqtri |
⊢ ( ( (,) ‘ 〈 0 , 0 〉 ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ |
213 |
212
|
olci |
⊢ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ 〈 0 , 0 〉 ) ∩ ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ∅ ) |
214 |
|
ineq1 |
⊢ ( ( (,) ‘ ( 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 |
⊢ ( ( (,) ‘ ( 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 |
⊢ ( ( (,) ‘ ( 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 |
⊢ ( ( (,) ‘ 〈 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 |
⊢ ( ( (,) ‘ 〈 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 |
⊢ ( ( (,) ‘ 〈 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 |
⊢ ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ 〈 0 , 0 〉 ) ) ∩ ( (,) ‘ 〈 0 , 0 〉 ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ 〈 0 , 0 〉 ) ) ∩ ∅ ) |
223 |
|
in0 |
⊢ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ 〈 0 , 0 〉 ) ) ∩ ∅ ) = ∅ |
224 |
222 223
|
eqtri |
⊢ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ 〈 0 , 0 〉 ) ) ∩ ( (,) ‘ 〈 0 , 0 〉 ) ) = ∅ |
225 |
224
|
olci |
⊢ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ 〈 0 , 0 〉 ) ) ∩ ( (,) ‘ 〈 0 , 0 〉 ) ) = ∅ ) |
226 |
|
ineq2 |
⊢ ( ( (,) ‘ ( 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 |
⊢ ( ( (,) ‘ ( 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 |
⊢ ( ( (,) ‘ ( 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 |
⊢ ( ( (,) ‘ 〈 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 |
⊢ ( ( (,) ‘ 〈 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 |
⊢ ( ( (,) ‘ 〈 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 |
⊢ ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( ∀ 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 |
⊢ ( 𝑧 = 𝑚 → ( 𝑧 ∈ ( 1 ... 𝑛 ) ↔ 𝑚 ∈ ( 1 ... 𝑛 ) ) ) |
237 |
|
fveq2 |
⊢ ( 𝑧 = 𝑚 → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑚 ) ) |
238 |
236 237
|
ifbieq1d |
⊢ ( 𝑧 = 𝑚 → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , 〈 0 , 0 〉 ) ) |
239 |
|
eqid |
⊢ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) |
240 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑚 ) ∈ V |
241 |
|
opex |
⊢ 〈 0 , 0 〉 ∈ V |
242 |
240 241
|
ifex |
⊢ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , 〈 0 , 0 〉 ) ∈ V |
243 |
238 239 242
|
fvmpt |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ‘ 𝑚 ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , 〈 0 , 0 〉 ) ) |
244 |
243
|
fveq2d |
⊢ ( 𝑚 ∈ ℕ → ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ‘ 𝑚 ) ) = ( (,) ‘ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , 〈 0 , 0 〉 ) ) ) |
245 |
|
fvif |
⊢ ( (,) ‘ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , 〈 0 , 0 〉 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ 〈 0 , 0 〉 ) ) |
246 |
244 245
|
eqtrdi |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 𝑚 = 𝑧 → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ 𝑧 ∈ ( 1 ... 𝑛 ) ) ) |
249 |
248 115
|
ifbieq1d |
⊢ ( 𝑚 = 𝑧 → if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑚 ) ) , ( (,) ‘ 〈 0 , 0 〉 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓 ‘ 𝑧 ) ) , ( (,) ‘ 〈 0 , 0 〉 ) ) ) |
250 |
249
|
disjor |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 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 |
⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) = seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) |
254 |
206 252 253
|
uniiccvol |
⊢ ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) , ℝ* , < ) ) |
255 |
254
|
adantr |
⊢ ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) , ℝ* , < ) ) |
256 |
|
rexpssxrxp |
⊢ ( ℝ × ℝ ) ⊆ ( ℝ* × ℝ* ) |
257 |
164 256
|
sstri |
⊢ { 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
|
sseldi |
⊢ ( ( 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 |
⊢ 0 ∈ ℝ* |
260 |
|
opelxpi |
⊢ ( ( 0 ∈ ℝ* ∧ 0 ∈ ℝ* ) → 〈 0 , 0 〉 ∈ ( ℝ* × ℝ* ) ) |
261 |
259 259 260
|
mp2an |
⊢ 〈 0 , 0 〉 ∈ ( ℝ* × ℝ* ) |
262 |
|
ifcl |
⊢ ( ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ* × ℝ* ) ∧ 〈 0 , 0 〉 ∈ ( ℝ* × ℝ* ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ∈ ( ℝ* × ℝ* ) ) |
263 |
258 261 262
|
sylancl |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ [,] : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ* |
266 |
265
|
a1i |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 𝑚 = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) → ( [,] ‘ 𝑚 ) = ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) |
269 |
263 264 267 268
|
fmptco |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℕ ) |
274 |
273 173
|
eleqtrdi |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
275 |
|
fzouzsplit |
⊢ ( ( 𝑛 + 1 ) ∈ ( ℤ≥ ‘ 1 ) → ( ℤ≥ ‘ 1 ) = ( ( 1 ..^ ( 𝑛 + 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) ) |
276 |
274 275
|
syl |
⊢ ( 𝑛 ∈ ℕ → ( ℤ≥ ‘ 1 ) = ( ( 1 ..^ ( 𝑛 + 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) ) |
277 |
173 276
|
syl5eq |
⊢ ( 𝑛 ∈ ℕ → ℕ = ( ( 1 ..^ ( 𝑛 + 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) ) |
278 |
|
nnz |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℤ ) |
279 |
|
fzval3 |
⊢ ( 𝑛 ∈ ℤ → ( 1 ... 𝑛 ) = ( 1 ..^ ( 𝑛 + 1 ) ) ) |
280 |
278 279
|
syl |
⊢ ( 𝑛 ∈ ℕ → ( 1 ... 𝑛 ) = ( 1 ..^ ( 𝑛 + 1 ) ) ) |
281 |
280
|
uneq1d |
⊢ ( 𝑛 ∈ ℕ → ( ( 1 ... 𝑛 ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = ( ( 1 ..^ ( 𝑛 + 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) ) |
282 |
277 281
|
eqtr4d |
⊢ ( 𝑛 ∈ ℕ → ℕ = ( ( 1 ... 𝑛 ) ∪ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) ) |
283 |
|
fvif |
⊢ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ 〈 0 , 0 〉 ) ) |
284 |
283
|
a1i |
⊢ ( 𝑛 ∈ ℕ → ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ 〈 0 , 0 〉 ) ) ) |
285 |
282 284
|
iuneq12d |
⊢ ( 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 |
⊢ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ∈ V |
287 |
286
|
dfiun3 |
⊢ ∪ 𝑧 ∈ ℕ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) = ∪ ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) |
288 |
|
iunxun |
⊢ ∪ 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 |
⊢ ( 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 |
⊢ ( 𝑧 ∈ ( 1 ... 𝑛 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ 〈 0 , 0 〉 ) ) = ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) |
291 |
290
|
iuneq2i |
⊢ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ 〈 0 , 0 〉 ) ) = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) |
292 |
291
|
a1i |
⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ 〈 0 , 0 〉 ) ) = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) |
293 |
|
uznfz |
⊢ ( 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) → ¬ 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ) |
294 |
293
|
adantl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → ¬ 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ) |
295 |
|
nncn |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ ) |
296 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
297 |
|
pncan |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑛 + 1 ) − 1 ) = 𝑛 ) |
298 |
295 296 297
|
sylancl |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑛 + 1 ) − 1 ) = 𝑛 ) |
299 |
298
|
oveq2d |
⊢ ( 𝑛 ∈ ℕ → ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) = ( 1 ... 𝑛 ) ) |
300 |
299
|
eleq2d |
⊢ ( 𝑛 ∈ ℕ → ( 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ↔ 𝑧 ∈ ( 1 ... 𝑛 ) ) ) |
301 |
300
|
notbid |
⊢ ( 𝑛 ∈ ℕ → ( ¬ 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ↔ ¬ 𝑧 ∈ ( 1 ... 𝑛 ) ) ) |
302 |
301
|
adantr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → ( ¬ 𝑧 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ↔ ¬ 𝑧 ∈ ( 1 ... 𝑛 ) ) ) |
303 |
294 302
|
mpbid |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → ¬ 𝑧 ∈ ( 1 ... 𝑛 ) ) |
304 |
303
|
iffalsed |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ 〈 0 , 0 〉 ) ) = ( [,] ‘ 〈 0 , 0 〉 ) ) |
305 |
304
|
iuneq2dv |
⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) , ( [,] ‘ 〈 0 , 0 〉 ) ) = ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ 〈 0 , 0 〉 ) ) |
306 |
292 305
|
uneq12d |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ < Or ℝ* |
311 |
310
|
a1i |
⊢ ( ( 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 |
⊢ ( 𝑛 ∈ ℕ ↔ 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) |
313 |
312
|
biimpi |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) |
314 |
313
|
adantl |
⊢ ( ( 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 |
⊢ ( 𝑢 ∈ ( 1 ... 𝑛 ) → 𝑢 ∈ ℕ ) |
316 |
172
|
ffvelrnda |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ) → ( 𝑢 + 𝑣 ) ∈ ℝ ) |
320 |
319
|
adantl |
⊢ ( ( ( 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 |
⊢ ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ ) |
322 |
321
|
rexrd |
⊢ ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ* ) |
323 |
|
eqidd |
⊢ ( 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 |
⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑚 ) , 〈 0 , 0 〉 ) = ( 𝑓 ‘ 𝑚 ) ) |
325 |
238 324
|
sylan9eqr |
⊢ ( ( 𝑚 ∈ ( 1 ... 𝑛 ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) = ( 𝑓 ‘ 𝑚 ) ) |
326 |
|
elfznn |
⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → 𝑚 ∈ ℕ ) |
327 |
240
|
a1i |
⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( 𝑓 ‘ 𝑚 ) ∈ V ) |
328 |
323 325 326 327
|
fvmptd |
⊢ ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ‘ 𝑚 ) = ( 𝑓 ‘ 𝑚 ) ) |
329 |
328
|
adantl |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑓 ‘ 𝑧 ) ∈ V |
332 |
331 241
|
ifex |
⊢ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ∈ V |
333 |
332 239
|
fnmpti |
⊢ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) Fn ℕ |
334 |
|
fvco2 |
⊢ ( ( ( 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 |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( ( 𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) ) |
339 |
337 326 338
|
syl2an |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
343 |
174
|
a1i |
⊢ ( 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
|
sseldi |
⊢ ( ( 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 |
⊢ 0 ∈ ℂ |
346 |
|
opelxpi |
⊢ ( ( 0 ∈ ℂ ∧ 0 ∈ ℂ ) → 〈 0 , 0 〉 ∈ ( ℂ × ℂ ) ) |
347 |
345 345 346
|
mp2an |
⊢ 〈 0 , 0 〉 ∈ ( ℂ × ℂ ) |
348 |
|
ifcl |
⊢ ( ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℂ × ℂ ) ∧ 〈 0 , 0 〉 ∈ ( ℂ × ℂ ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ∈ ( ℂ × ℂ ) ) |
349 |
344 347 348
|
sylancl |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( ( ( 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 |
⊢ ( 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
|
ffvelrnda |
⊢ ( ( 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 |
⊢ ( 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 ) } → seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) : ℕ ⟶ ℝ ) |
355 |
354
|
ffnd |
⊢ ( 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 ) } → seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) Fn ℕ ) |
356 |
|
fnfvelrn |
⊢ ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) Fn ℕ ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑛 ) ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ) |
357 |
355 356
|
sylan |
⊢ ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑛 ) ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ) |
358 |
342 357
|
eqeltrrd |
⊢ ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ) |
359 |
354
|
frnd |
⊢ ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ⊆ ℝ ) |
360 |
359
|
adantr |
⊢ ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ⊆ ℝ ) |
361 |
360
|
sselda |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ) → 𝑚 ∈ ℝ ) |
362 |
321
|
adantr |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ ) |
363 |
|
readdcl |
⊢ ( ( 𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ) → ( 𝑚 + 𝑢 ) ∈ ℝ ) |
364 |
363
|
adantl |
⊢ ( ( ( ( ( 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 |
⊢ ( 𝑚 ∈ ℝ → 𝑚 ∈ ℂ ) |
366 |
|
recn |
⊢ ( 𝑢 ∈ ℝ → 𝑢 ∈ ℂ ) |
367 |
|
recn |
⊢ ( 𝑣 ∈ ℝ → 𝑣 ∈ ℂ ) |
368 |
|
addass |
⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ ) → ( ( 𝑚 + 𝑢 ) + 𝑣 ) = ( 𝑚 + ( 𝑢 + 𝑣 ) ) ) |
369 |
365 366 367 368
|
syl3an |
⊢ ( ( 𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ) → ( ( 𝑚 + 𝑢 ) + 𝑣 ) = ( 𝑚 + ( 𝑢 + 𝑣 ) ) ) |
370 |
369
|
adantl |
⊢ ( ( ( ( ( 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 |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) → ( 𝑛 < 𝑡 ↔ ( 𝑛 + 1 ) ≤ 𝑡 ) ) |
372 |
371
|
biimpa |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( 𝑛 + 1 ) ≤ 𝑡 ) |
373 |
273
|
nnzd |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℤ ) |
374 |
|
nnz |
⊢ ( 𝑡 ∈ ℕ → 𝑡 ∈ ℤ ) |
375 |
|
eluz |
⊢ ( ( ( 𝑛 + 1 ) ∈ ℤ ∧ 𝑡 ∈ ℤ ) → ( 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑛 + 1 ) ≤ 𝑡 ) ) |
376 |
373 374 375
|
syl2an |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) → ( 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑛 + 1 ) ≤ 𝑡 ) ) |
377 |
376
|
adantr |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ↔ ( 𝑛 + 1 ) ≤ 𝑡 ) ) |
378 |
372 377
|
mpbird |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) |
379 |
378
|
adantlll |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ∈ ℕ ) |
383 |
381 382 353
|
syl2an |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) = ( ( seq 1 ( + , ( ( 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 |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
386 |
|
elfzelz |
⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → 𝑚 ∈ ℤ ) |
387 |
386
|
adantl |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℤ ) |
388 |
|
0red |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 ∈ ℝ ) |
389 |
273
|
nnred |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℝ ) |
390 |
389
|
ad3antrrr |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 + 1 ) ∈ ℝ ) |
391 |
386
|
zred |
⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → 𝑚 ∈ ℝ ) |
392 |
391
|
adantl |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℝ ) |
393 |
273
|
nngt0d |
⊢ ( 𝑛 ∈ ℕ → 0 < ( 𝑛 + 1 ) ) |
394 |
393
|
ad3antrrr |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 < ( 𝑛 + 1 ) ) |
395 |
|
elfzle1 |
⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → ( 𝑛 + 1 ) ≤ 𝑚 ) |
396 |
395
|
adantl |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 + 1 ) ≤ 𝑚 ) |
397 |
388 390 392 394 396
|
ltletrd |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 < 𝑚 ) |
398 |
|
elnnz |
⊢ ( 𝑚 ∈ ℕ ↔ ( 𝑚 ∈ ℤ ∧ 0 < 𝑚 ) ) |
399 |
387 397 398
|
sylanbrc |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℕ ) |
400 |
333 399 334
|
sylancr |
⊢ ( ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ ) |
403 |
402
|
adantr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑛 ∈ ℝ ) |
404 |
389
|
adantr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 + 1 ) ∈ ℝ ) |
405 |
391
|
adantl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℝ ) |
406 |
402
|
ltp1d |
⊢ ( 𝑛 ∈ ℕ → 𝑛 < ( 𝑛 + 1 ) ) |
407 |
406
|
adantr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑛 < ( 𝑛 + 1 ) ) |
408 |
395
|
adantl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 + 1 ) ≤ 𝑚 ) |
409 |
403 404 405 407 408
|
ltletrd |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑛 < 𝑚 ) |
410 |
409
|
adantr |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → 𝑛 < 𝑚 ) |
411 |
403 405
|
ltnled |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛 ) ) |
412 |
|
breq1 |
⊢ ( 𝑚 = 𝑧 → ( 𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛 ) ) |
413 |
412
|
equcoms |
⊢ ( 𝑧 = 𝑚 → ( 𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛 ) ) |
414 |
413
|
notbid |
⊢ ( 𝑧 = 𝑚 → ( ¬ 𝑚 ≤ 𝑛 ↔ ¬ 𝑧 ≤ 𝑛 ) ) |
415 |
411 414
|
sylan9bb |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ( 𝑛 < 𝑚 ↔ ¬ 𝑧 ≤ 𝑛 ) ) |
416 |
410 415
|
mpbid |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ¬ 𝑧 ≤ 𝑛 ) |
417 |
|
elfzle2 |
⊢ ( 𝑧 ∈ ( 1 ... 𝑛 ) → 𝑧 ≤ 𝑛 ) |
418 |
416 417
|
nsyl |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ¬ 𝑧 ∈ ( 1 ... 𝑛 ) ) |
419 |
418
|
iffalsed |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) = 〈 0 , 0 〉 ) |
420 |
386
|
adantl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℤ ) |
421 |
|
0red |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 ∈ ℝ ) |
422 |
393
|
adantr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 < ( 𝑛 + 1 ) ) |
423 |
421 404 405 422 408
|
ltletrd |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 0 < 𝑚 ) |
424 |
420 423 398
|
sylanbrc |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 𝑚 ∈ ℕ ) |
425 |
241
|
a1i |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → 〈 0 , 0 〉 ∈ V ) |
426 |
401 419 424 425
|
fvmptd |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ‘ 𝑚 ) = 〈 0 , 0 〉 ) |
427 |
426
|
ad4ant14 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( − : ( ℂ × ℂ ) ⟶ ℂ ∧ 〈 0 , 0 〉 ∈ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ‘ 〈 0 , 0 〉 ) = ( abs ‘ ( − ‘ 〈 0 , 0 〉 ) ) ) |
431 |
137 347 430
|
mp2an |
⊢ ( ( abs ∘ − ) ‘ 〈 0 , 0 〉 ) = ( abs ‘ ( − ‘ 〈 0 , 0 〉 ) ) |
432 |
|
df-ov |
⊢ ( 0 − 0 ) = ( − ‘ 〈 0 , 0 〉 ) |
433 |
|
0m0e0 |
⊢ ( 0 − 0 ) = 0 |
434 |
432 433
|
eqtr3i |
⊢ ( − ‘ 〈 0 , 0 〉 ) = 0 |
435 |
434
|
fveq2i |
⊢ ( abs ‘ ( − ‘ 〈 0 , 0 〉 ) ) = ( abs ‘ 0 ) |
436 |
|
abs0 |
⊢ ( abs ‘ 0 ) = 0 |
437 |
435 436
|
eqtri |
⊢ ( abs ‘ ( − ‘ 〈 0 , 0 〉 ) ) = 0 |
438 |
431 437
|
eqtri |
⊢ ( ( abs ∘ − ) ‘ 〈 0 , 0 〉 ) = 0 |
439 |
429 438
|
eqtrdi |
⊢ ( ( ( ( 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 |
⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → 𝑚 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) |
441 |
|
c0ex |
⊢ 0 ∈ V |
442 |
441
|
fvconst2 |
⊢ ( 𝑚 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) → ( ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ‘ 𝑚 ) = 0 ) |
443 |
440 442
|
syl |
⊢ ( 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) → ( ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ‘ 𝑚 ) = 0 ) |
444 |
443
|
adantl |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ‘ 𝑚 ) = 0 ) |
445 |
439 444
|
eqtr4d |
⊢ ( ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) = ( ℤ≥ ‘ ( 𝑛 + 1 ) ) |
448 |
447
|
ser0 |
⊢ ( 𝑡 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) → ( seq ( 𝑛 + 1 ) ( + , ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ) ‘ 𝑡 ) = 0 ) |
449 |
378 448
|
syl |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) → ( seq ( 𝑛 + 1 ) ( + , ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) × { 0 } ) ) ‘ 𝑡 ) = 0 ) |
450 |
446 449
|
eqtrd |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 1 ( + , ( ( 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 ) ) = ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) + 0 ) ) |
453 |
172
|
ffvelrnda |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( 𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ ) → ( 𝑚 + 𝑣 ) ∈ ℝ ) |
457 |
456
|
adantl |
⊢ ( ( ( 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 |
⊢ ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ ) |
459 |
458
|
ad2antrr |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ ) |
460 |
459
|
recnd |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℂ ) |
461 |
460
|
addid1d |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) + 0 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
462 |
452 461
|
eqtrd |
⊢ ( ( ( ( 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 1 ( + , ( ( 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 ) ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
463 |
384 462
|
eqtrd |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
464 |
453
|
ad5ant15 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ∈ ℝ ) |
467 |
466
|
leidd |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
468 |
463 467
|
eqbrtrd |
⊢ ( ( ( ( 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 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
469 |
|
elnnuz |
⊢ ( 𝑡 ∈ ℕ ↔ 𝑡 ∈ ( ℤ≥ ‘ 1 ) ) |
470 |
469
|
biimpi |
⊢ ( 𝑡 ∈ ℕ → 𝑡 ∈ ( ℤ≥ ‘ 1 ) ) |
471 |
470
|
ad2antlr |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( 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 |
⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 1 ≤ 𝑚 ) |
475 |
474
|
adantl |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 1 ≤ 𝑚 ) |
476 |
382
|
nnred |
⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ∈ ℝ ) |
477 |
476
|
adantl |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ∈ ℝ ) |
478 |
|
nnre |
⊢ ( 𝑡 ∈ ℕ → 𝑡 ∈ ℝ ) |
479 |
478
|
ad3antlr |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑡 ∈ ℝ ) |
480 |
402
|
ad3antrrr |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑛 ∈ ℝ ) |
481 |
|
elfzle2 |
⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ≤ 𝑡 ) |
482 |
481
|
adantl |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ≤ 𝑡 ) |
483 |
|
simplr |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑡 ≤ 𝑛 ) |
484 |
477 479 480 482 483
|
letrd |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ≤ 𝑛 ) |
485 |
|
elfzelz |
⊢ ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ∈ ℤ ) |
486 |
278
|
ad2antrr |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → 𝑛 ∈ ℤ ) |
487 |
|
elfz |
⊢ ( ( 𝑚 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛 ) ) ) |
488 |
174 487
|
mp3an2 |
⊢ ( ( 𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛 ) ) ) |
489 |
485 486 488
|
syl2anr |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛 ) ) ) |
490 |
475 484 489
|
mpbir2and |
⊢ ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ∈ ( 1 ... 𝑛 ) ) |
491 |
490
|
ad5ant2345 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( 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 |
⊢ ( 𝑧 ∈ ( 1 ... 𝑛 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) = ( 𝑓 ‘ 𝑧 ) ) |
495 |
493 494
|
syl |
⊢ ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑡 ) ) |
509 |
|
eluz |
⊢ ( ( 𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑡 ) ↔ 𝑡 ≤ 𝑛 ) ) |
510 |
374 278 509
|
syl2anr |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑡 ) ↔ 𝑡 ≤ 𝑛 ) ) |
511 |
510
|
biimpar |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡 ≤ 𝑛 ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑡 ) ) |
512 |
511
|
adantlll |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) → 𝑚 ∈ ℤ ) |
515 |
514
|
adantl |
⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℤ ) |
516 |
|
0red |
⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 0 ∈ ℝ ) |
517 |
|
peano2nn |
⊢ ( 𝑡 ∈ ℕ → ( 𝑡 + 1 ) ∈ ℕ ) |
518 |
517
|
nnred |
⊢ ( 𝑡 ∈ ℕ → ( 𝑡 + 1 ) ∈ ℝ ) |
519 |
518
|
adantr |
⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → ( 𝑡 + 1 ) ∈ ℝ ) |
520 |
514
|
zred |
⊢ ( 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) → 𝑚 ∈ ℝ ) |
521 |
520
|
adantl |
⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℝ ) |
522 |
517
|
nngt0d |
⊢ ( 𝑡 ∈ ℕ → 0 < ( 𝑡 + 1 ) ) |
523 |
522
|
adantr |
⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 0 < ( 𝑡 + 1 ) ) |
524 |
|
elfzle1 |
⊢ ( 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) → ( 𝑡 + 1 ) ≤ 𝑚 ) |
525 |
524
|
adantl |
⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → ( 𝑡 + 1 ) ≤ 𝑚 ) |
526 |
516 519 521 523 525
|
ltletrd |
⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 0 < 𝑚 ) |
527 |
515 526 398
|
sylanbrc |
⊢ ( ( 𝑡 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℕ ) |
528 |
527
|
adantlr |
⊢ ( ( ( 𝑡 ∈ ℕ ∧ 𝑡 ≤ 𝑛 ) ∧ 𝑚 ∈ ( ( 𝑡 + 1 ) ... 𝑛 ) ) → 𝑚 ∈ ℕ ) |
529 |
528
|
adantlll |
⊢ ( ( ( ( ( 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
|
ffvelrnda |
⊢ ( ( 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 |
|
ffvelrn |
⊢ ( ( − : ( ℂ × ℂ ) ⟶ ℂ ∧ ( 𝑓 ‘ 𝑚 ) ∈ ( ℂ × ℂ ) ) → ( − ‘ ( 𝑓 ‘ 𝑚 ) ) ∈ ℂ ) |
532 |
137 530 531
|
sylancr |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( − : ( ℂ × ℂ ) ⟶ ℂ ∧ ( 𝑓 ‘ 𝑚 ) ∈ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ‘ ( 𝑓 ‘ 𝑚 ) ) = ( abs ‘ ( − ‘ ( 𝑓 ‘ 𝑚 ) ) ) ) |
535 |
137 530 534
|
sylancr |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
541 |
508 540
|
eqbrtrd |
⊢ ( ( ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
542 |
402
|
ad2antlr |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
545 |
544
|
ralrimiva |
⊢ ( ( 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 ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
546 |
|
breq1 |
⊢ ( 𝑚 = ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) → ( 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) ) |
547 |
546
|
ralrn |
⊢ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) Fn ℕ → ( ∀ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) ) |
548 |
355 547
|
syl |
⊢ ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) ) |
549 |
548
|
adantr |
⊢ ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) ) |
550 |
545 549
|
mpbird |
⊢ ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
551 |
550
|
r19.21bi |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ) → 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
552 |
361 362 551
|
lensymd |
⊢ ( ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) ) → ¬ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) < 𝑚 ) |
553 |
311 322 358 552
|
supmax |
⊢ ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) , ℝ* , < ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
554 |
52 553
|
sylan |
⊢ ( ( 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 seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓 ‘ 𝑧 ) , 〈 0 , 0 〉 ) ) ) ) , ℝ* , < ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) |
555 |
255 309 554
|
3eqtr3rd |
⊢ ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) = ( vol* ‘ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∪ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ 〈 0 , 0 〉 ) ) ) ) |
556 |
|
elfznn |
⊢ ( 𝑧 ∈ ( 1 ... 𝑛 ) → 𝑧 ∈ ℕ ) |
557 |
164 65
|
sseldi |
⊢ ( ( 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 |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( 𝑓 ‘ 𝑧 ) = 〈 ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) 〉 ) |
559 |
558
|
fveq2d |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) = ( [,] ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) 〉 ) ) |
560 |
|
df-ov |
⊢ ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) = ( [,] ‘ 〈 ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) , ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) 〉 ) |
561 |
559 560
|
eqtr4di |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) = ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) |
562 |
|
xp1st |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ) |
563 |
|
xp2nd |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ) |
564 |
|
iccssre |
⊢ ( ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ⊆ ℝ ) |
565 |
562 563 564
|
syl2anc |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ⊆ ℝ ) |
566 |
561 565
|
eqsstrd |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ ) |
567 |
557 566
|
syl |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ ↔ ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ ℝ ) |
571 |
569 570
|
sylibr |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( ( 𝑛 + 1 ) ∈ ℤ → ( 𝑛 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) |
574 |
|
ne0i |
⊢ ( ( 𝑛 + 1 ) ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) → ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ≠ ∅ ) |
575 |
|
iunconst |
⊢ ( ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ≠ ∅ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ 〈 0 , 0 〉 ) = ( [,] ‘ 〈 0 , 0 〉 ) ) |
576 |
373 573 574 575
|
4syl |
⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ 〈 0 , 0 〉 ) = ( [,] ‘ 〈 0 , 0 〉 ) ) |
577 |
|
iccid |
⊢ ( 0 ∈ ℝ* → ( 0 [,] 0 ) = { 0 } ) |
578 |
259 577
|
ax-mp |
⊢ ( 0 [,] 0 ) = { 0 } |
579 |
|
df-ov |
⊢ ( 0 [,] 0 ) = ( [,] ‘ 〈 0 , 0 〉 ) |
580 |
578 579
|
eqtr3i |
⊢ { 0 } = ( [,] ‘ 〈 0 , 0 〉 ) |
581 |
576 580
|
eqtr4di |
⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ 〈 0 , 0 〉 ) = { 0 } ) |
582 |
|
snssi |
⊢ ( 0 ∈ ℝ → { 0 } ⊆ ℝ ) |
583 |
199 582
|
ax-mp |
⊢ { 0 } ⊆ ℝ |
584 |
581 583
|
eqsstrdi |
⊢ ( 𝑛 ∈ ℕ → ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ 〈 0 , 0 〉 ) ⊆ ℝ ) |
585 |
584
|
adantl |
⊢ ( ( 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 |
⊢ ( 𝑛 ∈ ℕ → ( vol* ‘ ∪ 𝑧 ∈ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ 〈 0 , 0 〉 ) ) = ( vol* ‘ { 0 } ) ) |
587 |
586
|
adantl |
⊢ ( ( 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 |
⊢ ( 0 ∈ ℝ → ( vol* ‘ { 0 } ) = 0 ) |
589 |
199 588
|
ax-mp |
⊢ ( vol* ‘ { 0 } ) = 0 |
590 |
587 589
|
eqtrdi |
⊢ ( ( 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 |
⊢ ( ( ∪ 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 |
⊢ ( ( 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 |
⊢ ( ( 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 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) = ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) |
594 |
593
|
breq2d |
⊢ ( ( 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 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) |
595 |
594
|
biimpd |
⊢ ( ( 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 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) → 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) |
596 |
595
|
reximdva |
⊢ ( 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 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) → ∃ 𝑛 ∈ ℕ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) |
597 |
596
|
adantl |
⊢ ( ( ( 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 < ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) → ∃ 𝑛 ∈ ℕ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) |
598 |
194 597
|
mpd |
⊢ ( ( ( 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 |
⊢ ( 1 ... 𝑛 ) ∈ Fin |
600 |
|
icccld |
⊢ ( ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ℝ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ) |
601 |
562 563 600
|
syl2anc |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( ( 1st ‘ ( 𝑓 ‘ 𝑧 ) ) [,] ( 2nd ‘ ( 𝑓 ‘ 𝑧 ) ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ) |
602 |
561 601
|
eqeltrd |
⊢ ( ( 𝑓 ‘ 𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ) |
603 |
557 602
|
syl |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ℝ = ∪ ( topGen ‘ ran (,) ) |
607 |
606
|
iuncld |
⊢ ( ( ( 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 |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑏 = ( 𝑓 ‘ 𝑧 ) → ( [,] ‘ 𝑏 ) = ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) |
611 |
610
|
sseq1d |
⊢ ( 𝑏 = ( 𝑓 ‘ 𝑧 ) → ( ( [,] ‘ 𝑏 ) ⊆ 𝐴 ↔ ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ) ) |
612 |
611
|
elrab |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ↔ ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ) |
618 |
616 617
|
sylibr |
⊢ ( 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( 𝑠 = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( 𝑠 ⊆ 𝐴 ↔ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ⊆ 𝐴 ) ) |
622 |
|
fveq2 |
⊢ ( 𝑠 = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( vol* ‘ 𝑠 ) = ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) |
623 |
622
|
breq2d |
⊢ ( 𝑠 = ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) → ( 𝑀 < ( vol* ‘ 𝑠 ) ↔ 𝑀 < ( vol* ‘ ∪ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓 ‘ 𝑧 ) ) ) ) ) |
624 |
621 623
|
anbi12d |
⊢ ( 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 |
⊢ ( ( ∪ 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 |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 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 |
⊢ ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) ) |
632 |
15 631
|
pm2.61dane |
⊢ ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠 ⊆ 𝐴 ∧ 𝑀 < ( vol* ‘ 𝑠 ) ) ) |