Metamath Proof Explorer


Theorem mblfinlem2

Description: Lemma for ismblfin , effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different definition of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018) (Revised by Brendan Leahy, 13-Jul-2018)

Ref Expression
Assertion mblfinlem2 ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( vol* ‘ 𝑠 ) ) )

Proof

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 = ∅ ) → ( ∅ ⊆ 𝐴𝑀 < ( vol* ‘ ∅ ) ) )
10 sseq1 ( 𝑠 = ∅ → ( 𝑠𝐴 ↔ ∅ ⊆ 𝐴 ) )
11 fveq2 ( 𝑠 = ∅ → ( vol* ‘ 𝑠 ) = ( vol* ‘ ∅ ) )
12 11 breq2d ( 𝑠 = ∅ → ( 𝑀 < ( vol* ‘ 𝑠 ) ↔ 𝑀 < ( vol* ‘ ∅ ) ) )
13 10 12 anbi12d ( 𝑠 = ∅ → ( ( 𝑠𝐴𝑀 < ( vol* ‘ 𝑠 ) ) ↔ ( ∅ ⊆ 𝐴𝑀 < ( vol* ‘ ∅ ) ) ) )
14 13 rspcev ( ( ∅ ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ∧ ( ∅ ⊆ 𝐴𝑀 < ( vol* ‘ ∅ ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( vol* ‘ 𝑠 ) ) )
15 3 9 14 sylancr ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 = ∅ ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( 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 = 𝑐𝑧 = 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 ) → ( 𝑓𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 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 ) → ( 𝑓𝑚 ) ∈ 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 ) ) → ( 𝑓𝑚 ) ∈ 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 ) → ( 𝑓𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 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 ) → ( 𝑓𝑧 ) ∈ 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 ) ) → ( 𝑓𝑧 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) )
68 35 dyaddisj ( ( ( 𝑓𝑚 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∧ ( 𝑓𝑧 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ) → ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
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 ) ) → ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
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 ) ) → ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
71 df-3or ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ↔ ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
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 ) ) → ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
73 elrabi ( ( 𝑓𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( 𝑓𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } )
74 fveq2 ( 𝑎 = ( 𝑓𝑚 ) → ( [,] ‘ 𝑎 ) = ( [,] ‘ ( 𝑓𝑚 ) ) )
75 74 sseq1d ( 𝑎 = ( 𝑓𝑚 ) → ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) ) )
76 eqeq1 ( 𝑎 = ( 𝑓𝑚 ) → ( 𝑎 = 𝑐 ↔ ( 𝑓𝑚 ) = 𝑐 ) )
77 75 76 imbi12d ( 𝑎 = ( 𝑓𝑚 ) → ( ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑚 ) = 𝑐 ) ) )
78 77 ralbidv ( 𝑎 = ( 𝑓𝑚 ) → ( ∀ 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 } ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑚 ) = 𝑐 ) ) )
79 78 elrab ( ( 𝑓𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ↔ ( ( 𝑓𝑚 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∧ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑚 ) = 𝑐 ) ) )
80 79 simprbi ( ( 𝑓𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 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 } ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑚 ) = 𝑐 ) )
81 fveq2 ( 𝑐 = ( 𝑓𝑧 ) → ( [,] ‘ 𝑐 ) = ( [,] ‘ ( 𝑓𝑧 ) ) )
82 81 sseq2d ( 𝑐 = ( 𝑓𝑧 ) → ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ) )
83 eqeq2 ( 𝑐 = ( 𝑓𝑧 ) → ( ( 𝑓𝑚 ) = 𝑐 ↔ ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) )
84 82 83 imbi12d ( 𝑐 = ( 𝑓𝑧 ) → ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑚 ) = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) → ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) ) )
85 84 rspcva ( ( ( 𝑓𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∧ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑚 ) = 𝑐 ) ) → ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) → ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) )
86 73 80 85 syl2anr ( ( ( 𝑓𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑓𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) → ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) )
87 elrabi ( ( 𝑓𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } → ( 𝑓𝑚 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } )
88 fveq2 ( 𝑎 = ( 𝑓𝑧 ) → ( [,] ‘ 𝑎 ) = ( [,] ‘ ( 𝑓𝑧 ) ) )
89 88 sseq1d ( 𝑎 = ( 𝑓𝑧 ) → ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) ) )
90 eqeq1 ( 𝑎 = ( 𝑓𝑧 ) → ( 𝑎 = 𝑐 ↔ ( 𝑓𝑧 ) = 𝑐 ) )
91 89 90 imbi12d ( 𝑎 = ( 𝑓𝑧 ) → ( ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑧 ) = 𝑐 ) ) )
92 91 ralbidv ( 𝑎 = ( 𝑓𝑧 ) → ( ∀ 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 } ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑧 ) = 𝑐 ) ) )
93 92 elrab ( ( 𝑓𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ↔ ( ( 𝑓𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∧ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑧 ) = 𝑐 ) ) )
94 93 simprbi ( ( 𝑓𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 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 } ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑧 ) = 𝑐 ) )
95 fveq2 ( 𝑐 = ( 𝑓𝑚 ) → ( [,] ‘ 𝑐 ) = ( [,] ‘ ( 𝑓𝑚 ) ) )
96 95 sseq2d ( 𝑐 = ( 𝑓𝑚 ) → ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) ↔ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) )
97 eqeq2 ( 𝑐 = ( 𝑓𝑚 ) → ( ( 𝑓𝑧 ) = 𝑐 ↔ ( 𝑓𝑧 ) = ( 𝑓𝑚 ) ) )
98 96 97 imbi12d ( 𝑐 = ( 𝑓𝑚 ) → ( ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑧 ) = 𝑐 ) ↔ ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) → ( 𝑓𝑧 ) = ( 𝑓𝑚 ) ) ) )
99 98 rspcva ( ( ( 𝑓𝑚 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∧ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ 𝑐 ) → ( 𝑓𝑧 ) = 𝑐 ) ) → ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) → ( 𝑓𝑧 ) = ( 𝑓𝑚 ) ) )
100 87 94 99 syl2an ( ( ( 𝑓𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑓𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) → ( 𝑓𝑧 ) = ( 𝑓𝑚 ) ) )
101 eqcom ( ( 𝑓𝑧 ) = ( 𝑓𝑚 ) ↔ ( 𝑓𝑚 ) = ( 𝑓𝑧 ) )
102 100 101 syl6ib ( ( ( 𝑓𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑓𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) → ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) )
103 86 102 jaod ( ( ( 𝑓𝑚 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ∧ ( 𝑓𝑧 ) ∈ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ∣ ∀ 𝑐 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ( ( [,] ‘ 𝑎 ) ⊆ ( [,] ‘ 𝑐 ) → 𝑎 = 𝑐 ) } ) → ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) → ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) )
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 ) ) → ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) → ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) )
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 ) ) → ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) → ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) )
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 ) ) → ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) → ( 𝑓𝑚 ) = ( 𝑓𝑧 ) ) )
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 ) ) → ( ( 𝑓𝑚 ) = ( 𝑓𝑧 ) → 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 ) ) → ( ( 𝑓𝑚 ) = ( 𝑓𝑧 ) → 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 ) ) → ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) → 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 ) ) → ( ( ( ( [,] ‘ ( 𝑓𝑚 ) ) ⊆ ( [,] ‘ ( 𝑓𝑧 ) ) ∨ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ( [,] ‘ ( 𝑓𝑚 ) ) ) ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) → ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ) )
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 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
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 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
114 eqeq1 ( 𝑚 = 𝑧 → ( 𝑚 = 𝑝𝑧 = 𝑝 ) )
115 2fveq3 ( 𝑚 = 𝑧 → ( (,) ‘ ( 𝑓𝑚 ) ) = ( (,) ‘ ( 𝑓𝑧 ) ) )
116 115 ineq1d ( 𝑚 = 𝑧 → ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ( ( (,) ‘ ( 𝑓𝑧 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) )
117 116 eqeq1d ( 𝑚 = 𝑧 → ( ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ↔ ( ( (,) ‘ ( 𝑓𝑧 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) )
118 114 117 orbi12d ( 𝑚 = 𝑧 → ( ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) ↔ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑧 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) ) )
119 118 ralbidv ( 𝑚 = 𝑧 → ( ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) ↔ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑧 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) ) )
120 119 cbvralvw ( ∀ 𝑚 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) ↔ ∀ 𝑧 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑧 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) )
121 eqeq2 ( 𝑧 = 𝑝 → ( 𝑚 = 𝑧𝑚 = 𝑝 ) )
122 2fveq3 ( 𝑧 = 𝑝 → ( (,) ‘ ( 𝑓𝑧 ) ) = ( (,) ‘ ( 𝑓𝑝 ) ) )
123 122 ineq2d ( 𝑧 = 𝑝 → ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) )
124 123 eqeq1d ( 𝑧 = 𝑝 → ( ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ↔ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) )
125 121 124 orbi12d ( 𝑧 = 𝑝 → ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) ) )
126 125 cbvralvw ( ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ↔ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) )
127 126 ralbii ( ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑚 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) )
128 122 disjor ( Disj 𝑧 ∈ ℕ ( (,) ‘ ( 𝑓𝑧 ) ) ↔ ∀ 𝑧 ∈ ℕ ∀ 𝑝 ∈ ℕ ( 𝑧 = 𝑝 ∨ ( ( (,) ‘ ( 𝑓𝑧 ) ) ∩ ( (,) ‘ ( 𝑓𝑝 ) ) ) = ∅ ) )
129 120 127 128 3bitr4ri ( Disj 𝑧 ∈ ℕ ( (,) ‘ ( 𝑓𝑧 ) ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
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 ( (,) ‘ ( 𝑓𝑧 ) ) )
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 ∈ ℕ0𝑦 ∈ 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 < 𝑧𝑀 < ( 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 ) → ( 𝑓𝑧 ) ∈ ( ≤ ∩ ( 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 ( ( ( 𝑓𝑧 ) ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ ⟨ 0 , 0 ⟩ ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ⟩ ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ( ∅ ∩ ( (,) ‘ ( 𝑓𝑧 ) ) )
211 0in ( ∅ ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅
212 210 211 eqtri ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅
213 212 olci ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ )
214 ineq1 ( ( (,) ‘ ( 𝑓𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) )
215 214 eqeq1d ( ( (,) ‘ ( 𝑓𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ↔ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
216 215 orbi2d ( ( (,) ‘ ( 𝑓𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ) )
217 ineq1 ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) )
218 217 eqeq1d ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ↔ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
219 218 orbi2d ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ) )
220 216 219 ifboth ( ( ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ( 𝑓𝑚 ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ∧ ( 𝑚 = 𝑧 ∨ ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ) → ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
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 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) )
222 209 ineq2i ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ∅ )
223 in0 ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ∅ ) = ∅
224 222 223 eqtri ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅
225 224 olci ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅ )
226 ineq2 ( ( (,) ‘ ( 𝑓𝑧 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) )
227 226 eqeq1d ( ( (,) ‘ ( 𝑓𝑧 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ↔ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
228 227 orbi2d ( ( (,) ‘ ( 𝑓𝑧 ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ↔ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) ) )
229 ineq2 ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) )
230 229 eqeq1d ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅ ↔ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
231 230 orbi2d ( ( (,) ‘ ⟨ 0 , 0 ⟩ ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅ ) ↔ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) ) )
232 228 231 ifboth ( ( ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ( 𝑓𝑧 ) ) ) = ∅ ) ∧ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = ∅ ) ) → ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 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 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 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 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
235 disjeq2 ( ∀ 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) → ( Disj 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) ↔ Disj 𝑚 ∈ ℕ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) )
236 eleq1w ( 𝑧 = 𝑚 → ( 𝑧 ∈ ( 1 ... 𝑛 ) ↔ 𝑚 ∈ ( 1 ... 𝑛 ) ) )
237 fveq2 ( 𝑧 = 𝑚 → ( 𝑓𝑧 ) = ( 𝑓𝑚 ) )
238 236 237 ifbieq1d ( 𝑧 = 𝑚 → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑚 ) , ⟨ 0 , 0 ⟩ ) )
239 eqid ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) )
240 fvex ( 𝑓𝑚 ) ∈ V
241 opex ⟨ 0 , 0 ⟩ ∈ V
242 240 241 ifex if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑚 ) , ⟨ 0 , 0 ⟩ ) ∈ V
243 238 239 242 fvmpt ( 𝑚 ∈ ℕ → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑚 ) , ⟨ 0 , 0 ⟩ ) )
244 243 fveq2d ( 𝑚 ∈ ℕ → ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = ( (,) ‘ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑚 ) , ⟨ 0 , 0 ⟩ ) ) )
245 fvif ( (,) ‘ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑚 ) , ⟨ 0 , 0 ⟩ ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) )
246 244 245 eqtrdi ( 𝑚 ∈ ℕ → ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) )
247 235 246 mprg ( Disj 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) ↔ Disj 𝑚 ∈ ℕ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) )
248 eleq1w ( 𝑚 = 𝑧 → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ 𝑧 ∈ ( 1 ... 𝑛 ) ) )
249 248 115 ifbieq1d ( 𝑚 = 𝑧 → if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) )
250 249 disjor ( Disj 𝑚 ∈ ℕ if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ) = ∅ ) )
251 247 250 bitri ( Disj 𝑚 ∈ ℕ ( (,) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) ↔ ∀ 𝑚 ∈ ℕ ∀ 𝑧 ∈ ℕ ( 𝑚 = 𝑧 ∨ ( if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑚 ) ) , ( (,) ‘ ⟨ 0 , 0 ⟩ ) ) ∩ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( (,) ‘ ( 𝑓𝑧 ) ) , ( (,) ‘ ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
253 eqid seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) = seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) → ( 𝑓𝑧 ) ∈ ( 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 ( ( ( 𝑓𝑧 ) ∈ ( ℝ* × ℝ* ) ∧ ⟨ 0 , 0 ⟩ ∈ ( ℝ* × ℝ* ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) → ( [,] ‘ 𝑚 ) = ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) )
284 283 a1i ( 𝑛 ∈ ℕ → ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
285 282 284 iuneq12d ( 𝑛 ∈ ℕ → 𝑧 ∈ ℕ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = 𝑧 ∈ ( ( 1 ... 𝑛 ) ∪ ( ℤ ‘ ( 𝑛 + 1 ) ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
286 fvex ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ∈ V
287 286 dfiun3 𝑧 ∈ ℕ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) )
288 iunxun 𝑧 ∈ ( ( 1 ... 𝑛 ) ∪ ( ℤ ‘ ( 𝑛 + 1 ) ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ∪ 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
289 285 287 288 3eqtr3g ( 𝑛 ∈ ℕ → ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ∪ 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) )
290 iftrue ( 𝑧 ∈ ( 1 ... 𝑛 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ( [,] ‘ ( 𝑓𝑧 ) ) )
291 290 iuneq2i 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) )
292 291 a1i ( 𝑛 ∈ ℕ → 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) )
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 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = ( [,] ‘ ⟨ 0 , 0 ⟩ ) )
305 304 iuneq2dv ( 𝑛 ∈ ℕ → 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) )
306 292 305 uneq12d ( 𝑛 ∈ ℕ → ( 𝑧 ∈ ( 1 ... 𝑛 ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ∪ 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( [,] ‘ ( 𝑓𝑧 ) ) , ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) = ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∪ 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) )
307 289 306 eqtrd ( 𝑛 ∈ ℕ → ran ( 𝑧 ∈ ℕ ↦ ( [,] ‘ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∪ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) = ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∪ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) = ( vol* ‘ ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∪ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) )
324 iftrue ( 𝑚 ∈ ( 1 ... 𝑛 ) → if ( 𝑚 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑚 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓𝑚 ) )
325 238 324 sylan9eqr ( ( 𝑚 ∈ ( 1 ... 𝑛 ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓𝑚 ) )
326 elfznn ( 𝑚 ∈ ( 1 ... 𝑛 ) → 𝑚 ∈ ℕ )
327 240 a1i ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( 𝑓𝑚 ) ∈ V )
328 323 325 326 327 fvmptd ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ( 𝑓𝑚 ) )
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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ( 𝑓𝑚 ) )
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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = ( ( abs ∘ − ) ‘ ( 𝑓𝑚 ) ) )
331 fvex ( 𝑓𝑧 ) ∈ V
332 331 241 ifex if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ∈ V
333 332 239 fnmpti ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) Fn ℕ
334 fvco2 ( ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) Fn ℕ ∧ 𝑚 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
335 333 326 334 sylancr ( 𝑚 ∈ ( 1 ... 𝑛 ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ∘ − ) ‘ ( 𝑓𝑚 ) ) )
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 ∘ − ) ‘ ( 𝑓𝑚 ) ) )
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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) → ( 𝑓𝑧 ) ∈ ( 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 ( ( ( 𝑓𝑧 ) ∈ ( ℂ × ℂ ) ∧ ⟨ 0 , 0 ⟩ ∈ ( ℂ × ℂ ) ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) : ℕ ⟶ ( ℂ × ℂ ) )
351 fco ( ( ( abs ∘ − ) : ( ℂ × ℂ ) ⟶ ℝ ∧ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) : ℕ ⟶ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) Fn ℕ )
356 fnfvelrn ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) Fn ℕ ∧ 𝑛 ∈ ℕ ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) = ( ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) + ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) )
401 eqidd ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ↔ ¬ 𝑚𝑛 ) )
412 breq1 ( 𝑚 = 𝑧 → ( 𝑚𝑛𝑧𝑛 ) )
413 412 equcoms ( 𝑧 = 𝑚 → ( 𝑚𝑛𝑧𝑛 ) )
414 413 notbid ( 𝑧 = 𝑚 → ( ¬ 𝑚𝑛 ↔ ¬ 𝑧𝑛 ) )
415 411 414 sylan9bb ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ( 𝑛 < 𝑚 ↔ ¬ 𝑧𝑛 ) )
416 410 415 mpbid ( ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ¬ 𝑧𝑛 )
417 elfzle2 ( 𝑧 ∈ ( 1 ... 𝑛 ) → 𝑧𝑛 )
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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ⟨ 0 , 0 ⟩ )
427 426 ad4ant14 ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑛 < 𝑡 ) ∧ 𝑚 ∈ ( ( 𝑛 + 1 ) ... 𝑡 ) ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) + ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑛 ) + ( seq ( 𝑛 + 1 ) ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) ∧ 𝑡𝑛 ) → 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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) = ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → 𝑧 = 𝑚 )
474 elfzle1 ( 𝑚 ∈ ( 1 ... 𝑡 ) → 1 ≤ 𝑚 )
475 474 adantl ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 1 ≤ 𝑚 )
476 382 nnred ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ∈ ℝ )
477 476 adantl ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚 ∈ ℝ )
478 nnre ( 𝑡 ∈ ℕ → 𝑡 ∈ ℝ )
479 478 ad3antlr ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑡 ∈ ℝ )
480 402 ad3antrrr ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑛 ∈ ℝ )
481 elfzle2 ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚𝑡 )
482 481 adantl ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚𝑡 )
483 simplr ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑡𝑛 )
484 477 479 480 482 483 letrd ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → 𝑚𝑛 )
485 elfzelz ( 𝑚 ∈ ( 1 ... 𝑡 ) → 𝑚 ∈ ℤ )
486 278 ad2antrr ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) → 𝑛 ∈ ℤ )
487 elfz ( ( 𝑚 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚𝑚𝑛 ) ) )
488 174 487 mp3an2 ( ( 𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚𝑚𝑛 ) ) )
489 485 486 488 syl2anr ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( 𝑚 ∈ ( 1 ... 𝑛 ) ↔ ( 1 ≤ 𝑚𝑚𝑛 ) ) )
490 475 484 489 mpbir2and ( ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) ∧ 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 ) ∧ 𝑡𝑛 ) ∧ 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 ) ∧ 𝑡𝑛 ) ∧ 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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → 𝑧 ∈ ( 1 ... 𝑛 ) )
494 iftrue ( 𝑧 ∈ ( 1 ... 𝑛 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓𝑧 ) )
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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓𝑧 ) )
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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → ( 𝑓𝑧 ) = ( 𝑓𝑚 ) )
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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) ∧ 𝑧 = 𝑚 ) → if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) = ( 𝑓𝑚 ) )
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 ) ∧ 𝑡𝑛 ) ∧ 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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( 𝑓𝑚 ) ∈ 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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) = ( 𝑓𝑚 ) )
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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ‘ 𝑚 ) ) = ( ( abs ∘ − ) ‘ ( 𝑓𝑚 ) ) )
502 333 382 334 sylancr ( 𝑚 ∈ ( 1 ... 𝑡 ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) ∧ 𝑡𝑛 ) → 𝑓 : ℕ ⟶ { 𝑎 ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 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 ∘ − ) ‘ ( 𝑓𝑚 ) ) )
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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ 𝑓 ) ‘ 𝑚 ) = ( ( abs ∘ − ) ‘ ( 𝑓𝑚 ) ) )
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 ) ∧ 𝑡𝑛 ) ∧ 𝑚 ∈ ( 1 ... 𝑡 ) ) → ( ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) ∧ 𝑡𝑛 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) = ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑡 ) )
509 eluz ( ( 𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → ( 𝑛 ∈ ( ℤ𝑡 ) ↔ 𝑡𝑛 ) )
510 374 278 509 syl2anr ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) → ( 𝑛 ∈ ( ℤ𝑡 ) ↔ 𝑡𝑛 ) )
511 510 biimpar ( ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ ) ∧ 𝑡𝑛 ) → 𝑛 ∈ ( ℤ𝑡 ) )
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 ) ∧ 𝑡𝑛 ) → 𝑛 ∈ ( ℤ𝑡 ) )
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 ) ∧ 𝑡𝑛 ) ∧ 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 ∧ 𝑡𝑛 ) ∧ 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 ) ∧ 𝑡𝑛 ) ∧ 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 ) → ( 𝑓𝑚 ) ∈ ( ℂ × ℂ ) )
531 ffvelrn ( ( − : ( ℂ × ℂ ) ⟶ ℂ ∧ ( 𝑓𝑚 ) ∈ ( ℂ × ℂ ) ) → ( − ‘ ( 𝑓𝑚 ) ) ∈ 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 ) → ( − ‘ ( 𝑓𝑚 ) ) ∈ 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 ‘ ( − ‘ ( 𝑓𝑚 ) ) ) )
534 fvco3 ( ( − : ( ℂ × ℂ ) ⟶ ℂ ∧ ( 𝑓𝑚 ) ∈ ( ℂ × ℂ ) ) → ( ( abs ∘ − ) ‘ ( 𝑓𝑚 ) ) = ( abs ‘ ( − ‘ ( 𝑓𝑚 ) ) ) )
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 ∘ − ) ‘ ( 𝑓𝑚 ) ) = ( abs ‘ ( − ‘ ( 𝑓𝑚 ) ) ) )
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 ‘ ( − ‘ ( 𝑓𝑚 ) ) ) )
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 ) ∧ 𝑡𝑛 ) ∧ 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 ) ∧ 𝑡𝑛 ) ∧ 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 ) ∧ 𝑡𝑛 ) → ( 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 ) ∧ 𝑡𝑛 ) → ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) )
546 breq1 ( 𝑚 = ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) → ( 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) ‘ 𝑡 ) ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ) )
547 546 ralrn ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) Fn ℕ → ( ∀ 𝑚 ∈ ran seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 0 , 0 ⟩ ) ) ) ) 𝑚 ≤ ( seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) ‘ 𝑛 ) ↔ ∀ 𝑡 ∈ ℕ ( seq 1 ( + , ( ( abs ∘ − ) ∘ ( 𝑧 ∈ ℕ ↦ if ( 𝑧 ∈ ( 1 ... 𝑛 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) , ( 𝑓𝑧 ) , ⟨ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∪ 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 ) → ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) )
558 1st2nd2 ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( 𝑓𝑧 ) = ⟨ ( 1st ‘ ( 𝑓𝑧 ) ) , ( 2nd ‘ ( 𝑓𝑧 ) ) ⟩ )
559 558 fveq2d ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓𝑧 ) ) = ( [,] ‘ ⟨ ( 1st ‘ ( 𝑓𝑧 ) ) , ( 2nd ‘ ( 𝑓𝑧 ) ) ⟩ ) )
560 df-ov ( ( 1st ‘ ( 𝑓𝑧 ) ) [,] ( 2nd ‘ ( 𝑓𝑧 ) ) ) = ( [,] ‘ ⟨ ( 1st ‘ ( 𝑓𝑧 ) ) , ( 2nd ‘ ( 𝑓𝑧 ) ) ⟩ )
561 559 560 eqtr4di ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓𝑧 ) ) = ( ( 1st ‘ ( 𝑓𝑧 ) ) [,] ( 2nd ‘ ( 𝑓𝑧 ) ) ) )
562 xp1st ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝑓𝑧 ) ) ∈ ℝ )
563 xp2nd ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝑓𝑧 ) ) ∈ ℝ )
564 iccssre ( ( ( 1st ‘ ( 𝑓𝑧 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓𝑧 ) ) ∈ ℝ ) → ( ( 1st ‘ ( 𝑓𝑧 ) ) [,] ( 2nd ‘ ( 𝑓𝑧 ) ) ) ⊆ ℝ )
565 562 563 564 syl2anc ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( ( 1st ‘ ( 𝑓𝑧 ) ) [,] ( 2nd ‘ ( 𝑓𝑧 ) ) ) ⊆ ℝ )
566 561 565 eqsstrd ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) → ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ) → ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ℝ )
570 iunss ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ℝ ↔ ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ ℝ ∧ 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ⊆ ℝ ∧ ( vol* ‘ 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) = 0 ) → ( vol* ‘ ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∪ 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) = ( vol* ‘ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) )
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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∪ 𝑧 ∈ ( ℤ ‘ ( 𝑛 + 1 ) ) ( [,] ‘ ⟨ 0 , 0 ⟩ ) ) ) = ( vol* ‘ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) )
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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) )
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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) )
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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) )
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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) )
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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) )
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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) )
599 fzfi ( 1 ... 𝑛 ) ∈ Fin
600 icccld ( ( ( 1st ‘ ( 𝑓𝑧 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝑓𝑧 ) ) ∈ ℝ ) → ( ( 1st ‘ ( 𝑓𝑧 ) ) [,] ( 2nd ‘ ( 𝑓𝑧 ) ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
601 562 563 600 syl2anc ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( ( 1st ‘ ( 𝑓𝑧 ) ) [,] ( 2nd ‘ ( 𝑓𝑧 ) ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
602 561 601 eqeltrd ( ( 𝑓𝑧 ) ∈ ( ℝ × ℝ ) → ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( 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 ) → ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( 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 ) ) → ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
606 uniretop ℝ = ( topGen ‘ ran (,) )
607 606 iuncld ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( 1 ... 𝑛 ) ∈ Fin ∧ ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ) → 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) ) → 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) )
610 fveq2 ( 𝑏 = ( 𝑓𝑧 ) → ( [,] ‘ 𝑏 ) = ( [,] ‘ ( 𝑓𝑧 ) ) )
611 610 sseq1d ( 𝑏 = ( 𝑓𝑧 ) → ( ( [,] ‘ 𝑏 ) ⊆ 𝐴 ↔ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 𝐴 ) )
612 611 elrab ( ( 𝑓𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } ↔ ( ( 𝑓𝑧 ) ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∧ ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 𝐴 ) )
613 612 simprbi ( ( 𝑓𝑧 ) ∈ { 𝑏 ∈ ran ( 𝑥 ∈ ℤ , 𝑦 ∈ ℕ0 ↦ ⟨ ( 𝑥 / ( 2 ↑ 𝑦 ) ) , ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑦 ) ) ⟩ ) ∣ ( [,] ‘ 𝑏 ) ⊆ 𝐴 } → ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) → ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ) → ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 𝐴 )
617 iunss ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 𝐴 ↔ ∀ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) ) → 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) ) → 𝑀 < ( vol* ‘ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) )
621 sseq1 ( 𝑠 = 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) → ( 𝑠𝐴 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 𝐴 ) )
622 fveq2 ( 𝑠 = 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) → ( vol* ‘ 𝑠 ) = ( vol* ‘ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) )
623 622 breq2d ( 𝑠 = 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) → ( 𝑀 < ( vol* ‘ 𝑠 ) ↔ 𝑀 < ( vol* ‘ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) )
624 621 623 anbi12d ( 𝑠 = 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) → ( ( 𝑠𝐴𝑀 < ( vol* ‘ 𝑠 ) ) ↔ ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 𝐴𝑀 < ( vol* ‘ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) ) )
625 624 rspcev ( ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ∧ ( 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ⊆ 𝐴𝑀 < ( vol* ‘ 𝑧 ∈ ( 1 ... 𝑛 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( 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 ) ( [,] ‘ ( 𝑓𝑧 ) ) ) ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( 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 (,) ) ) ( 𝑠𝐴𝑀 < ( 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 (,) ) ) ( 𝑠𝐴𝑀 < ( vol* ‘ 𝑠 ) ) )
631 17 630 exlimddv ( ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) ∧ 𝐴 ≠ ∅ ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( vol* ‘ 𝑠 ) ) )
632 15 631 pm2.61dane ( ( 𝐴 ∈ ( topGen ‘ ran (,) ) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < ( vol* ‘ 𝐴 ) ) → ∃ 𝑠 ∈ ( Clsd ‘ ( topGen ‘ ran (,) ) ) ( 𝑠𝐴𝑀 < ( vol* ‘ 𝑠 ) ) )