Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mblfinlem2 Structured version   Visualization version   GIF version

Theorem mblfinlem2 37282
Description: Lemma for ismblfin 37285, 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.)
Assertion
Ref Expression
mblfinlem2 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
Distinct variable groups:   𝐴,𝑠   𝑀,𝑠

Proof of Theorem mblfinlem2
Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑚 𝑛 𝑝 𝑡 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 retop 24739 . . . 4 (topGen‘ran (,)) ∈ Top
2 0cld 23003 . . . 4 ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
31, 2ax-mp 5 . . 3 ∅ ∈ (Clsd‘(topGen‘ran (,)))
4 simpl3 1190 . . . . 5 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘𝐴))
5 fveq2 6896 . . . . . 6 (𝐴 = ∅ → (vol*‘𝐴) = (vol*‘∅))
65adantl 480 . . . . 5 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (vol*‘𝐴) = (vol*‘∅))
74, 6breqtrd 5175 . . . 4 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘∅))
8 0ss 4398 . . . 4 ∅ ⊆ 𝐴
97, 8jctil 518 . . 3 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (∅ ⊆ 𝐴𝑀 < (vol*‘∅)))
10 sseq1 4002 . . . . 5 (𝑠 = ∅ → (𝑠𝐴 ↔ ∅ ⊆ 𝐴))
11 fveq2 6896 . . . . . 6 (𝑠 = ∅ → (vol*‘𝑠) = (vol*‘∅))
1211breq2d 5161 . . . . 5 (𝑠 = ∅ → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∅)))
1310, 12anbi12d 630 . . . 4 (𝑠 = ∅ → ((𝑠𝐴𝑀 < (vol*‘𝑠)) ↔ (∅ ⊆ 𝐴𝑀 < (vol*‘∅))))
1413rspcev 3606 . . 3 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝐴𝑀 < (vol*‘∅))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
153, 9, 14sylancr 585 . 2 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
16 mblfinlem1 37281 . . . 4 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
17163ad2antl1 1182 . . 3 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
18 simpl3 1190 . . . . . . . . 9 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑀 < (vol*‘𝐴))
19 f1ofo 6845 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
20 rnco2 6259 . . . . . . . . . . . . . . . . 17 ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
21 forn 6813 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran 𝑓 = {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
2221imaeq2d 6064 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] “ ran 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
2320, 22eqtrid 2777 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
2423unieqd 4922 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
2519, 24syl 17 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
2625adantl 480 . . . . . . . . . . . . 13 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 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 7426 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦)))
28 oveq1 7426 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1))
2928oveq1d 7434 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦)))
3027, 29opeq12d 4883 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑢 → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩)
31 oveq2 7427 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣))
3231oveq2d 7435 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣)))
3331oveq2d 7435 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣)))
3432, 33opeq12d 4883 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 → ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))⟩)
3530, 34cbvmpov 7515 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) = (𝑢 ∈ ℤ, 𝑣 ∈ ℕ0 ↦ ⟨(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))⟩)
36 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧))
3736sseq1d 4008 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐)))
38 eqeq1 2729 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑧 → (𝑎 = 𝑐𝑧 = 𝑐))
3937, 38imbi12d 343 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
4039ralbidv 3167 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑧 → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
4140cbvrabv 3429 . . . . . . . . . . . . . . 15 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} = {𝑧 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)}
42 ssrab2 4073 . . . . . . . . . . . . . . . 16 {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
4342a1i 11 . . . . . . . . . . . . . . 15 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
4435, 41, 43dyadmbllem 25589 . . . . . . . . . . . . . 14 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
4544adantr 479 . . . . . . . . . . . . 13 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
4626, 45eqtr4d 2768 . . . . . . . . . . . 12 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ran ([,] ∘ 𝑓) = ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}))
47 opnmbllem0 37280 . . . . . . . . . . . . . 14 (𝐴 ∈ (topGen‘ran (,)) → ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
48473ad2ant1 1130 . . . . . . . . . . . . 13 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
4948adantr 479 . . . . . . . . . . . 12 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
5046, 49eqtrd 2765 . . . . . . . . . . 11 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ran ([,] ∘ 𝑓) = 𝐴)
5150fveq2d 6900 . . . . . . . . . 10 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘ ran ([,] ∘ 𝑓)) = (vol*‘𝐴))
52 f1of 6838 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
53 ssrab2 4073 . . . . . . . . . . . . . 14 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}
5435dyadf 25581 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
55 frn 6730 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ ( ≤ ∩ (ℝ × ℝ)))
5654, 55ax-mp 5 . . . . . . . . . . . . . . 15 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ ( ≤ ∩ (ℝ × ℝ))
5742, 56sstri 3986 . . . . . . . . . . . . . 14 {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ( ≤ ∩ (ℝ × ℝ))
5853, 57sstri 3986 . . . . . . . . . . . . 13 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ( ≤ ∩ (ℝ × ℝ))
59 fss 6739 . . . . . . . . . . . . 13 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ( ≤ ∩ (ℝ × ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
6052, 58, 59sylancl 584 . . . . . . . . . . . 12 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
6153, 42sstri 3986 . . . . . . . . . . . . . . . . . . . 20 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
62 ffvelcdm 7090 . . . . . . . . . . . . . . . . . . . 20 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (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)})
6361, 62sselid 3974 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
6463adantrr 715 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑓𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
65 ffvelcdm 7090 . . . . . . . . . . . . . . . . . . . 20 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (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)})
6661, 65sselid 3974 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
6766adantrl 714 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑓𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
6835dyaddisj 25586 . . . . . . . . . . . . . . . . . 18 (((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))) = ∅))
6964, 67, 68syl2anc 582 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚)) ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
7052, 69sylan 578 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚)) ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
71 df-3or 1085 . . . . . . . . . . . . . . . 16 ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚)) ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))) ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
7270, 71sylib 217 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))) ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
73 elrabi 3673 . . . . . . . . . . . . . . . . . . . . . 22 ((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 6896 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = (𝑓𝑚) → ([,]‘𝑎) = ([,]‘(𝑓𝑚)))
7574sseq1d 4008 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = (𝑓𝑚) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓𝑚)) ⊆ ([,]‘𝑐)))
76 eqeq1 2729 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = (𝑓𝑚) → (𝑎 = 𝑐 ↔ (𝑓𝑚) = 𝑐))
7775, 76imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = (𝑓𝑚) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓𝑚)) ⊆ ([,]‘𝑐) → (𝑓𝑚) = 𝑐)))
7877ralbidv 3167 . . . . . . . . . . . . . . . . . . . . . . . 24 (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)))
7978elrab 3679 . . . . . . . . . . . . . . . . . . . . . . 23 ((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)))
8079simprbi 495 . . . . . . . . . . . . . . . . . . . . . 22 ((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 6896 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (𝑓𝑧) → ([,]‘𝑐) = ([,]‘(𝑓𝑧)))
8281sseq2d 4009 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑓𝑧) → (([,]‘(𝑓𝑚)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧))))
83 eqeq2 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑓𝑧) → ((𝑓𝑚) = 𝑐 ↔ (𝑓𝑚) = (𝑓𝑧)))
8482, 83imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = (𝑓𝑧) → ((([,]‘(𝑓𝑚)) ⊆ ([,]‘𝑐) → (𝑓𝑚) = 𝑐) ↔ (([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) → (𝑓𝑚) = (𝑓𝑧))))
8584rspcva 3604 . . . . . . . . . . . . . . . . . . . . . 22 (((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)))
8673, 80, 85syl2anr 595 . . . . . . . . . . . . . . . . . . . . 21 (((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 3673 . . . . . . . . . . . . . . . . . . . . . . 23 ((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 6896 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = (𝑓𝑧) → ([,]‘𝑎) = ([,]‘(𝑓𝑧)))
8988sseq1d 4008 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = (𝑓𝑧) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓𝑧)) ⊆ ([,]‘𝑐)))
90 eqeq1 2729 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = (𝑓𝑧) → (𝑎 = 𝑐 ↔ (𝑓𝑧) = 𝑐))
9189, 90imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = (𝑓𝑧) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓𝑧)) ⊆ ([,]‘𝑐) → (𝑓𝑧) = 𝑐)))
9291ralbidv 3167 . . . . . . . . . . . . . . . . . . . . . . . . 25 (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)))
9392elrab 3679 . . . . . . . . . . . . . . . . . . . . . . . 24 ((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)))
9493simprbi 495 . . . . . . . . . . . . . . . . . . . . . . 23 ((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 6896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = (𝑓𝑚) → ([,]‘𝑐) = ([,]‘(𝑓𝑚)))
9695sseq2d 4009 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (𝑓𝑚) → (([,]‘(𝑓𝑧)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))))
97 eqeq2 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (𝑓𝑚) → ((𝑓𝑧) = 𝑐 ↔ (𝑓𝑧) = (𝑓𝑚)))
9896, 97imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑓𝑚) → ((([,]‘(𝑓𝑧)) ⊆ ([,]‘𝑐) → (𝑓𝑧) = 𝑐) ↔ (([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚)) → (𝑓𝑧) = (𝑓𝑚))))
9998rspcva 3604 . . . . . . . . . . . . . . . . . . . . . . 23 (((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)))
10087, 94, 99syl2an 594 . . . . . . . . . . . . . . . . . . . . . 22 (((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 2732 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑧) = (𝑓𝑚) ↔ (𝑓𝑚) = (𝑓𝑧))
102100, 101imbitrdi 250 . . . . . . . . . . . . . . . . . . . . 21 (((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)))
10386, 102jaod 857 . . . . . . . . . . . . . . . . . . . 20 (((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)))
10462, 65, 103syl2an 594 . . . . . . . . . . . . . . . . . . 19 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) ∧ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))) → (𝑓𝑚) = (𝑓𝑧)))
105104anandis 676 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))) → (𝑓𝑚) = (𝑓𝑧)))
10652, 105sylan 578 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))) → (𝑓𝑚) = (𝑓𝑧)))
107 f1of1 6837 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ–1-1→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
108 f1veqaeq 7267 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ–1-1→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑓𝑚) = (𝑓𝑧) → 𝑚 = 𝑧))
109107, 108sylan 578 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑓𝑚) = (𝑓𝑧) → 𝑚 = 𝑧))
110106, 109syld 47 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))) → 𝑚 = 𝑧))
111110orim1d 963 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))) ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) → (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅)))
11272, 111mpd 15 . . . . . . . . . . . . . 14 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
113112ralrimivva 3190 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
114 eqeq1 2729 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 → (𝑚 = 𝑝𝑧 = 𝑝))
115 2fveq3 6901 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑧 → ((,)‘(𝑓𝑚)) = ((,)‘(𝑓𝑧)))
116115ineq1d 4209 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑧 → (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))))
117116eqeq1d 2727 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 → ((((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅ ↔ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅))
118114, 117orbi12d 916 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑧 → ((𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅) ↔ (𝑧 = 𝑝 ∨ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅)))
119118ralbidv 3167 . . . . . . . . . . . . . . 15 (𝑚 = 𝑧 → (∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅)))
120119cbvralvw 3224 . . . . . . . . . . . . . 14 (∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅))
121 eqeq2 2737 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑝 → (𝑚 = 𝑧𝑚 = 𝑝))
122 2fveq3 6901 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑝 → ((,)‘(𝑓𝑧)) = ((,)‘(𝑓𝑝)))
123122ineq2d 4210 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑝 → (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))))
124123eqeq1d 2727 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑝 → ((((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅ ↔ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅))
125121, 124orbi12d 916 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑝 → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅)))
126125cbvralvw 3224 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅))
127126ralbii 3082 . . . . . . . . . . . . . 14 (∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ ∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅))
128122disjor 5129 . . . . . . . . . . . . . 14 (Disj 𝑧 ∈ ℕ ((,)‘(𝑓𝑧)) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅))
129120, 127, 1283bitr4ri 303 . . . . . . . . . . . . 13 (Disj 𝑧 ∈ ℕ ((,)‘(𝑓𝑧)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
130113, 129sylibr 233 . . . . . . . . . . . 12 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → Disj 𝑧 ∈ ℕ ((,)‘(𝑓𝑧)))
131 eqid 2725 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
13260, 130, 131uniiccvol 25570 . . . . . . . . . . 11 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (vol*‘ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
133132adantl 480 . . . . . . . . . 10 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
13451, 133eqtr3d 2767 . . . . . . . . 9 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘𝐴) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
13518, 134breqtrd 5175 . . . . . . . 8 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑀 < sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
136 absf 15328 . . . . . . . . . . . 12 abs:ℂ⟶ℝ
137 subf 11499 . . . . . . . . . . . 12 − :(ℂ × ℂ)⟶ℂ
138 fco 6747 . . . . . . . . . . . 12 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
139136, 137, 138mp2an 690 . . . . . . . . . . 11 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
140 zre 12600 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
141 2re 12324 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
142 reexpcl 14084 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ ℝ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℝ)
143141, 142mpan 688 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℝ)
144 2cn 12325 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℂ
145 2ne0 12354 . . . . . . . . . . . . . . . . . . . . 21 2 ≠ 0
146 nn0z 12621 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℕ0𝑦 ∈ ℤ)
147 expne0i 14100 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0)
148144, 145, 146, 147mp3an12i 1461 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → (2↑𝑦) ≠ 0)
149143, 148jca 510 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0))
150 redivcl 11971 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → (𝑥 / (2↑𝑦)) ∈ ℝ)
151 peano2re 11424 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
152 redivcl 11971 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
153151, 152syl3an1 1160 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
154150, 153opelxpd 5717 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
1551543expb 1117 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0)) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
156140, 149, 155syl2an 594 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
157156rgen2 3187 . . . . . . . . . . . . . . . . 17 𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ)
158 eqid 2725 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
159158fmpo 8073 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ) ↔ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ))
160157, 159mpbi 229 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ)
161 frn 6730 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℝ × ℝ))
162160, 161ax-mp 5 . . . . . . . . . . . . . . 15 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℝ × ℝ)
16342, 162sstri 3986 . . . . . . . . . . . . . 14 {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ × ℝ)
16453, 163sstri 3986 . . . . . . . . . . . . 13 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ × ℝ)
165 ax-resscn 11202 . . . . . . . . . . . . . 14 ℝ ⊆ ℂ
166 xpss12 5693 . . . . . . . . . . . . . 14 ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
167165, 165, 166mp2an 690 . . . . . . . . . . . . 13 (ℝ × ℝ) ⊆ (ℂ × ℂ)
168164, 167sstri 3986 . . . . . . . . . . . 12 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℂ × ℂ)
169 fss 6739 . . . . . . . . . . . 12 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℂ × ℂ)) → 𝑓:ℕ⟶(ℂ × ℂ))
170168, 169mpan2 689 . . . . . . . . . . 11 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶(ℂ × ℂ))
171 fco 6747 . . . . . . . . . . 11 (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ)
172139, 170, 171sylancr 585 . . . . . . . . . 10 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ)
173 nnuz 12903 . . . . . . . . . . 11 ℕ = (ℤ‘1)
174 1z 12630 . . . . . . . . . . . 12 1 ∈ ℤ
175174a1i 11 . . . . . . . . . . 11 (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → 1 ∈ ℤ)
176 ffvelcdm 7090 . . . . . . . . . . 11 ((((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑛) ∈ ℝ)
177173, 175, 176serfre 14037 . . . . . . . . . 10 (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ)
178 frn 6730 . . . . . . . . . . 11 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ)
179 ressxr 11295 . . . . . . . . . . 11 ℝ ⊆ ℝ*
180178, 179sstrdi 3989 . . . . . . . . . 10 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
18152, 172, 177, 1804syl 19 . . . . . . . . 9 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
182 rexr 11297 . . . . . . . . . 10 (𝑀 ∈ ℝ → 𝑀 ∈ ℝ*)
1831823ad2ant2 1131 . . . . . . . . 9 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → 𝑀 ∈ ℝ*)
184 supxrlub 13344 . . . . . . . . 9 ((ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*𝑀 ∈ ℝ*) → (𝑀 < sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ ∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧))
185181, 183, 184syl2anr 595 . . . . . . . 8 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (𝑀 < sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ ∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧))
186135, 185mpbid 231 . . . . . . 7 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧)
187 seqfn 14019 . . . . . . . . . 10 (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ‘1))
188174, 187ax-mp 5 . . . . . . . . 9 seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ‘1)
189173fneq2i 6653 . . . . . . . . 9 (seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ‘1))
190188, 189mpbir 230 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ
191 breq2 5153 . . . . . . . . 9 (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → (𝑀 < 𝑧𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
192191rexrn 7096 . . . . . . . 8 (seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ → (∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧 ↔ ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
193190, 192ax-mp 5 . . . . . . 7 (∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧 ↔ ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
194186, 193sylib 217 . . . . . 6 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
19560ffvelcdmda 7093 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
196 0le0 12351 . . . . . . . . . . . . . . . . . 18 0 ≤ 0
197 df-br 5150 . . . . . . . . . . . . . . . . . 18 (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
198196, 197mpbi 229 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ ≤
199 0re 11253 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
200 opelxpi 5715 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
201199, 199, 200mp2an 690 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ (ℝ × ℝ)
202 elin 3960 . . . . . . . . . . . . . . . . 17 (⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨0, 0⟩ ∈ ≤ ∧ ⟨0, 0⟩ ∈ (ℝ × ℝ)))
203198, 201, 202mpbir2an 709 . . . . . . . . . . . . . . . 16 ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
204 ifcl 4575 . . . . . . . . . . . . . . . 16 (((𝑓𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
205195, 203, 204sylancl 584 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
206205fmpttd 7124 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
207 df-ov 7422 . . . . . . . . . . . . . . . . . . . . . 22 (0(,)0) = ((,)‘⟨0, 0⟩)
208 iooid 13392 . . . . . . . . . . . . . . . . . . . . . 22 (0(,)0) = ∅
209207, 208eqtr3i 2755 . . . . . . . . . . . . . . . . . . . . 21 ((,)‘⟨0, 0⟩) = ∅
210209ineq1i 4206 . . . . . . . . . . . . . . . . . . . 20 (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = (∅ ∩ ((,)‘(𝑓𝑧)))
211 0in 4395 . . . . . . . . . . . . . . . . . . . 20 (∅ ∩ ((,)‘(𝑓𝑧))) = ∅
212210, 211eqtri 2753 . . . . . . . . . . . . . . . . . . 19 (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅
213212olci 864 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅)
214 ineq1 4203 . . . . . . . . . . . . . . . . . . . . 21 (((,)‘(𝑓𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))))
215214eqeq1d 2727 . . . . . . . . . . . . . . . . . . . 20 (((,)‘(𝑓𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → ((((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅))
216215orbi2d 913 . . . . . . . . . . . . . . . . . . 19 (((,)‘(𝑓𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅)))
217 ineq1 4203 . . . . . . . . . . . . . . . . . . . . 21 (((,)‘⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))))
218217eqeq1d 2727 . . . . . . . . . . . . . . . . . . . 20 (((,)‘⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → ((((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅))
219218orbi2d 913 . . . . . . . . . . . . . . . . . . 19 (((,)‘⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → ((𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅)))
220216, 219ifboth 4569 . . . . . . . . . . . . . . . . . 18 (((𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ∧ (𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅))
221112, 213, 220sylancl 584 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅))
222209ineq2i 4207 . . . . . . . . . . . . . . . . . . 19 (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅)
223 in0 4393 . . . . . . . . . . . . . . . . . . 19 (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅) = ∅
224222, 223eqtri 2753 . . . . . . . . . . . . . . . . . 18 (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅
225224olci 864 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅)
226 ineq2 4204 . . . . . . . . . . . . . . . . . . . 20 (((,)‘(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⟩))))
227226eqeq1d 2727 . . . . . . . . . . . . . . . . . . 19 (((,)‘(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⟩))) = ∅))
228227orbi2d 913 . . . . . . . . . . . . . . . . . 18 (((,)‘(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 4204 . . . . . . . . . . . . . . . . . . . 20 (((,)‘⟨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⟩))))
230229eqeq1d 2727 . . . . . . . . . . . . . . . . . . 19 (((,)‘⟨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⟩))) = ∅))
231230orbi2d 913 . . . . . . . . . . . . . . . . . 18 (((,)‘⟨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⟩))) = ∅)))
232228, 231ifboth 4569 . . . . . . . . . . . . . . . . 17 (((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⟩))) = ∅))
233221, 225, 232sylancl 584 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
234233ralrimivva 3190 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
235 disjeq2 5118 . . . . . . . . . . . . . . . . 17 (∀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 2808 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑚 → (𝑧 ∈ (1...𝑛) ↔ 𝑚 ∈ (1...𝑛)))
237 fveq2 6896 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑚 → (𝑓𝑧) = (𝑓𝑚))
238236, 237ifbieq1d 4554 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑚 → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩))
239 eqid 2725 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))
240 fvex 6909 . . . . . . . . . . . . . . . . . . . . 21 (𝑓𝑚) ∈ V
241 opex 5466 . . . . . . . . . . . . . . . . . . . . 21 ⟨0, 0⟩ ∈ V
242240, 241ifex 4580 . . . . . . . . . . . . . . . . . . . 20 if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩) ∈ V
243238, 239, 242fvmpt 7004 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩))
244243fveq2d 6900 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ → ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)) = ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩)))
245 fvif 6912 . . . . . . . . . . . . . . . . . 18 ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩))
246244, 245eqtrdi 2781 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ → ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)))
247235, 246mprg 3056 . . . . . . . . . . . . . . . 16 (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 2808 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑧 → (𝑚 ∈ (1...𝑛) ↔ 𝑧 ∈ (1...𝑛)))
249248, 115ifbieq1d 4554 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 → if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓𝑧)), ((,)‘⟨0, 0⟩)))
250249disjor 5129 . . . . . . . . . . . . . . . 16 (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⟩))) = ∅))
251247, 250bitri 274 . . . . . . . . . . . . . . 15 (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⟩))) = ∅))
252234, 251sylibr 233 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → Disj 𝑚 ∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)))
253 eqid 2725 . . . . . . . . . . . . . 14 seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) = seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))
254206, 252, 253uniiccvol 25570 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (vol*‘ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))), ℝ*, < ))
255254adantr 479 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))), ℝ*, < ))
256 rexpssxrxp 11296 . . . . . . . . . . . . . . . . . . . . 21 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
257164, 256sstri 3986 . . . . . . . . . . . . . . . . . . . 20 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ* × ℝ*)
258257, 65sselid 3974 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓𝑧) ∈ (ℝ* × ℝ*))
259 0xr 11298 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ*
260 opelxpi 5715 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ⟨0, 0⟩ ∈ (ℝ* × ℝ*))
261259, 259, 260mp2an 690 . . . . . . . . . . . . . . . . . . 19 ⟨0, 0⟩ ∈ (ℝ* × ℝ*)
262 ifcl 4575 . . . . . . . . . . . . . . . . . . 19 (((𝑓𝑧) ∈ (ℝ* × ℝ*) ∧ ⟨0, 0⟩ ∈ (ℝ* × ℝ*)) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ (ℝ* × ℝ*))
263258, 261, 262sylancl 584 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ (ℝ* × ℝ*))
264 eqidd 2726 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))
265 iccf 13465 . . . . . . . . . . . . . . . . . . . 20 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
266265a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*)
267266feqmptd 6966 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → [,] = (𝑚 ∈ (ℝ* × ℝ*) ↦ ([,]‘𝑚)))
268 fveq2 6896 . . . . . . . . . . . . . . . . . 18 (𝑚 = if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) → ([,]‘𝑚) = ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))
269263, 264, 267, 268fmptco 7138 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))) = (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))
27052, 269syl 17 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))) = (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))
271270rneqd 5940 . . . . . . . . . . . . . . 15 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))) = ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))
272271unieqd 4922 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))) = ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))
273 peano2nn 12262 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
274273, 173eleqtrdi 2835 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ (ℤ‘1))
275 fzouzsplit 13707 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 + 1) ∈ (ℤ‘1) → (ℤ‘1) = ((1..^(𝑛 + 1)) ∪ (ℤ‘(𝑛 + 1))))
276274, 275syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (ℤ‘1) = ((1..^(𝑛 + 1)) ∪ (ℤ‘(𝑛 + 1))))
277173, 276eqtrid 2777 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ℕ = ((1..^(𝑛 + 1)) ∪ (ℤ‘(𝑛 + 1))))
278 nnz 12617 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
279 fzval3 13741 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℤ → (1...𝑛) = (1..^(𝑛 + 1)))
280278, 279syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (1...𝑛) = (1..^(𝑛 + 1)))
281280uneq1d 4159 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((1...𝑛) ∪ (ℤ‘(𝑛 + 1))) = ((1..^(𝑛 + 1)) ∪ (ℤ‘(𝑛 + 1))))
282277, 281eqtr4d 2768 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ℕ = ((1...𝑛) ∪ (ℤ‘(𝑛 + 1))))
283 fvif 6912 . . . . . . . . . . . . . . . . . 18 ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩))
284283a1i 11 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)))
285282, 284iuneq12d 5025 . . . . . . . . . . . . . . . 16 (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 6909 . . . . . . . . . . . . . . . . 17 ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) ∈ V
287286dfiun3 5969 . . . . . . . . . . . . . . . 16 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))
288 iunxun 5098 . . . . . . . . . . . . . . . 16 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⟩)))
289285, 287, 2883eqtr3g 2788 . . . . . . . . . . . . . . 15 (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 4536 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘(𝑓𝑧)))
291290iuneq2i 5018 . . . . . . . . . . . . . . . . 17 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)) = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))
292291a1i 11 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)) = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)))
293 uznfz 13624 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (ℤ‘(𝑛 + 1)) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
294293adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
295 nncn 12258 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
296 ax-1cn 11203 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℂ
297 pncan 11503 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
298295, 296, 297sylancl 584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛)
299298oveq2d 7435 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ℕ → (1...((𝑛 + 1) − 1)) = (1...𝑛))
300299eleq2d 2811 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → (𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ 𝑧 ∈ (1...𝑛)))
301300notbid 317 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
302301adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ‘(𝑛 + 1))) → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
303294, 302mpbid 231 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...𝑛))
304303iffalsed 4541 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ‘(𝑛 + 1))) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘⟨0, 0⟩))
305304iuneq2dv 5021 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑧 ∈ (ℤ‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)) = 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩))
306292, 305uneq12d 4161 . . . . . . . . . . . . . . 15 (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⟩)))
307289, 306eqtrd 2765 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))) = ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∪ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩)))
308272, 307sylan9eq 2785 . . . . . . . . . . . . 13 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))) = ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∪ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩)))
309308fveq2d 6900 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) = (vol*‘( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∪ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩))))
310 xrltso 13160 . . . . . . . . . . . . . . 15 < Or ℝ*
311310a1i 11 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → < Or ℝ*)
312 elnnuz 12904 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
313312biimpi 215 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
314313adantl 480 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ‘1))
315 elfznn 13570 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (1...𝑛) → 𝑢 ∈ ℕ)
316172ffvelcdmda 7093 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑢 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑢) ∈ ℝ)
317315, 316sylan2 591 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑢 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑢) ∈ ℝ)
318317adantlr 713 . . . . . . . . . . . . . . . 16 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑢 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑢) ∈ ℝ)
319 readdcl 11228 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ)
320319adantl 480 . . . . . . . . . . . . . . . 16 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ (𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑢 + 𝑣) ∈ ℝ)
321314, 318, 320seqcl 14028 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
322321rexrd 11301 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ*)
323 eqidd 2726 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (1...𝑛) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))
324 iftrue 4536 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (1...𝑛) → if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩) = (𝑓𝑚))
325238, 324sylan9eqr 2787 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ (1...𝑛) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = (𝑓𝑚))
326 elfznn 13570 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (1...𝑛) → 𝑚 ∈ ℕ)
327240a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (1...𝑛) → (𝑓𝑚) ∈ V)
328323, 325, 326, 327fvmptd 7011 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (1...𝑛) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓𝑚))
329328adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓𝑚))
330329fveq2d 6900 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)) = ((abs ∘ − )‘(𝑓𝑚)))
331 fvex 6909 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑧) ∈ V
332331, 241ifex 4580 . . . . . . . . . . . . . . . . . . . . 21 if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ V
333332, 239fnmpti 6699 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) Fn ℕ
334 fvco2 6994 . . . . . . . . . . . . . . . . . . . 20 (((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)))
335333, 326, 334sylancr 585 . . . . . . . . . . . . . . . . . . 19 (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)))
336335adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))‘𝑚) = ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)))
337 ffn 6723 . . . . . . . . . . . . . . . . . . 19 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓 Fn ℕ)
338 fvco2 6994 . . . . . . . . . . . . . . . . . . 19 ((𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓𝑚)))
339337, 326, 338syl2an 594 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓𝑚)))
340330, 336, 3393eqtr4d 2775 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))‘𝑚) = (((abs ∘ − ) ∘ 𝑓)‘𝑚))
341340adantlr 713 . . . . . . . . . . . . . . . 16 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))‘𝑚) = (((abs ∘ − ) ∘ 𝑓)‘𝑚))
342314, 341seqfveq 14032 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
343174a1i 11 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 1 ∈ ℤ)
344168, 65sselid 3974 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓𝑧) ∈ (ℂ × ℂ))
345 0cn 11243 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
346 opelxpi 5715 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨0, 0⟩ ∈ (ℂ × ℂ))
347345, 345, 346mp2an 690 . . . . . . . . . . . . . . . . . . . . . 22 ⟨0, 0⟩ ∈ (ℂ × ℂ)
348 ifcl 4575 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓𝑧) ∈ (ℂ × ℂ) ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ (ℂ × ℂ))
349344, 347, 348sylancl 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ (ℂ × ℂ))
350349fmpttd 7124 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)):ℕ⟶(ℂ × ℂ))
351 fco 6747 . . . . . . . . . . . . . . . . . . . 20 (((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)
352139, 350, 351sylancr 585 . . . . . . . . . . . . . . . . . . 19 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))):ℕ⟶ℝ)
353352ffvelcdmda 7093 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))‘𝑚) ∈ ℝ)
354173, 343, 353serfre 14037 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))):ℕ⟶ℝ)
355354ffnd 6724 . . . . . . . . . . . . . . . 16 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) Fn ℕ)
356 fnfvelrn 7089 . . . . . . . . . . . . . . . 16 ((seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) Fn ℕ ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑛) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))))
357355, 356sylan 578 . . . . . . . . . . . . . . 15 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑛) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))))
358342, 357eqeltrrd 2826 . . . . . . . . . . . . . 14 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))))
359354frnd 6731 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) ⊆ ℝ)
360359adantr 479 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) ⊆ ℝ)
361360sselda 3976 . . . . . . . . . . . . . . 15 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))) → 𝑚 ∈ ℝ)
362321adantr 479 . . . . . . . . . . . . . . 15 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
363 readdcl 11228 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑚 + 𝑢) ∈ ℝ)
364363adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ (𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ)) → (𝑚 + 𝑢) ∈ ℝ)
365 recn 11235 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℝ → 𝑚 ∈ ℂ)
366 recn 11235 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
367 recn 11235 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ ℝ → 𝑣 ∈ ℂ)
368 addass 11232 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
369365, 366, 367, 368syl3an 1157 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
370369adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ (𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
371 nnltp1le 12656 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 < 𝑡 ↔ (𝑛 + 1) ≤ 𝑡))
372371biimpa 475 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑛 + 1) ≤ 𝑡)
373273nnzd 12623 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℤ)
374 nnz 12617 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ ℕ → 𝑡 ∈ ℤ)
375 eluz 12874 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑛 + 1) ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ∈ (ℤ‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
376373, 374, 375syl2an 594 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (ℤ‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
377376adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑡 ∈ (ℤ‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
378372, 377mpbird 256 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ‘(𝑛 + 1)))
379378adantlll 716 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ‘(𝑛 + 1)))
380313ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑛 ∈ (ℤ‘1))
381 simplll 773 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
382 elfznn 13570 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℕ)
383381, 382, 353syl2an 594 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))‘𝑚) ∈ ℝ)
384364, 370, 379, 380, 383seqsplit 14041 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) = ((seq1( + , ((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)))
385342ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
386 elfzelz 13541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℤ)
387386adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
388 0red 11254 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
389273nnred 12265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
390389ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
391386zred 12704 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℝ)
392391adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
393273nngt0d 12299 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑛 ∈ ℕ → 0 < (𝑛 + 1))
394393ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
395 elfzle1 13544 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 ∈ ((𝑛 + 1)...𝑡) → (𝑛 + 1) ≤ 𝑚)
396395adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
397388, 390, 392, 394, 396ltletrd 11411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
398 elnnz 12606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 ∈ ℕ ↔ (𝑚 ∈ ℤ ∧ 0 < 𝑚))
399387, 397, 398sylanbrc 581 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ)
400333, 399, 334sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((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 2726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((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 12257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
403402adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 ∈ ℝ)
404389adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
405391adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
406402ltp1d 12182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
407406adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < (𝑛 + 1))
408395adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
409403, 404, 405, 407, 408ltletrd 11411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < 𝑚)
410409adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → 𝑛 < 𝑚)
411403, 405ltnled 11398 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 < 𝑚 ↔ ¬ 𝑚𝑛))
412 breq1 5152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑚 = 𝑧 → (𝑚𝑛𝑧𝑛))
413412equcoms 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑧 = 𝑚 → (𝑚𝑛𝑧𝑛))
414413notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 = 𝑚 → (¬ 𝑚𝑛 ↔ ¬ 𝑧𝑛))
415411, 414sylan9bb 508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → (𝑛 < 𝑚 ↔ ¬ 𝑧𝑛))
416410, 415mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧𝑛)
417 elfzle2 13545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 ∈ (1...𝑛) → 𝑧𝑛)
418416, 417nsyl 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ∈ (1...𝑛))
419418iffalsed 4541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = ⟨0, 0⟩)
420386adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
421 0red 11254 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
422393adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
423421, 404, 405, 422, 408ltletrd 11411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
424420, 423, 398sylanbrc 581 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ)
425241a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ⟨0, 0⟩ ∈ V)
426401, 419, 424, 425fvmptd 7011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = ⟨0, 0⟩)
427426ad4ant14 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = ⟨0, 0⟩)
428427fveq2d 6900 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)) = ((abs ∘ − )‘⟨0, 0⟩))
429400, 428eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((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 6996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (( − :(ℂ × ℂ)⟶ℂ ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩)))
431137, 347, 430mp2an 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩))
432 df-ov 7422 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 − 0) = ( − ‘⟨0, 0⟩)
433 0m0e0 12370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 − 0) = 0
434432, 433eqtr3i 2755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ( − ‘⟨0, 0⟩) = 0
435434fveq2i 6899 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (abs‘( − ‘⟨0, 0⟩)) = (abs‘0)
436 abs0 15276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (abs‘0) = 0
437435, 436eqtri 2753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (abs‘( − ‘⟨0, 0⟩)) = 0
438431, 437eqtri 2753 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((abs ∘ − )‘⟨0, 0⟩) = 0
439429, 438eqtrdi 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))‘𝑚) = 0)
440 elfzuz 13537 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ (ℤ‘(𝑛 + 1)))
441 c0ex 11245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ V
442441fvconst2 7216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 ∈ (ℤ‘(𝑛 + 1)) → (((ℤ‘(𝑛 + 1)) × {0})‘𝑚) = 0)
443440, 442syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 ∈ ((𝑛 + 1)...𝑡) → (((ℤ‘(𝑛 + 1)) × {0})‘𝑚) = 0)
444443adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((ℤ‘(𝑛 + 1)) × {0})‘𝑚) = 0)
445439, 444eqtr4d 2768 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((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))
446378, 445seqfveq 14032 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((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 2725 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑛 + 1))
448447ser0 14060 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ (ℤ‘(𝑛 + 1)) → (seq(𝑛 + 1)( + , ((ℤ‘(𝑛 + 1)) × {0}))‘𝑡) = 0)
449378, 448syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((ℤ‘(𝑛 + 1)) × {0}))‘𝑡) = 0)
450446, 449eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) = 0)
451450adantlll 716 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) = 0)
452385, 451oveq12d 7437 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡)) = ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) + 0))
453172ffvelcdmda 7093 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
454326, 453sylan2 591 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
455454adantlr 713 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
456 readdcl 11228 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑚 + 𝑣) ∈ ℝ)
457456adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑚 + 𝑣) ∈ ℝ)
458314, 455, 457seqcl 14028 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
459458ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
460459recnd 11279 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℂ)
461460addridd 11451 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) + 0) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
462452, 461eqtrd 2765 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡)) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
463384, 462eqtrd 2765 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
464453ad5ant15 757 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
465326, 464sylan2 591 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
466380, 465, 364seqcl 14028 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
467466leidd 11817 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
468463, 467eqbrtrd 5171 . . . . . . . . . . . . . . . . . . 19 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
469 elnnuz 12904 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 ∈ ℕ ↔ 𝑡 ∈ (ℤ‘1))
470469biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 ∈ ℕ → 𝑡 ∈ (ℤ‘1))
471470ad2antlr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → 𝑡 ∈ (ℤ‘1))
472 eqidd 2726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))
473 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 = 𝑚)
474 elfzle1 13544 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 ∈ (1...𝑡) → 1 ≤ 𝑚)
475474adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 1 ≤ 𝑚)
476382nnred 12265 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℝ)
477476adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℝ)
478 nnre 12257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 ∈ ℕ → 𝑡 ∈ ℝ)
479478ad3antlr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ∈ ℝ)
480402ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑛 ∈ ℝ)
481 elfzle2 13545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 ∈ (1...𝑡) → 𝑚𝑡)
482481adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚𝑡)
483 simplr 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡𝑛)
484477, 479, 480, 482, 483letrd 11408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚𝑛)
485 elfzelz 13541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℤ)
486278ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → 𝑛 ∈ ℤ)
487 elfz 13530 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑚 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚𝑚𝑛)))
488174, 487mp3an2 1445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚𝑚𝑛)))
489485, 486, 488syl2anr 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚𝑚𝑛)))
490475, 484, 489mpbir2and 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛))
491490ad5ant2345 1367 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛))
492491adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑚 ∈ (1...𝑛))
493473, 492eqeltrd 2825 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 ∈ (1...𝑛))
494 iftrue 4536 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = (𝑓𝑧))
495493, 494syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = (𝑓𝑧))
496237adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → (𝑓𝑧) = (𝑓𝑚))
497495, 496eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = (𝑓𝑚))
498382adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℕ)
499240a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑓𝑚) ∈ V)
500472, 497, 498, 499fvmptd 7011 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓𝑚))
501500fveq2d 6900 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)) = ((abs ∘ − )‘(𝑓𝑚)))
502333, 382, 334sylancr 585 . . . . . . . . . . . . . . . . . . . . . . 23 (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)))
503502adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 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 773 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 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 6996 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓𝑚)))
506504, 382, 505syl2an 594 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓𝑚)))
507501, 503, 5063eqtr4d 2775 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))‘𝑚) = (((abs ∘ − ) ∘ 𝑓)‘𝑚))
508471, 507seqfveq 14032 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑡))
509 eluz 12874 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ𝑡) ↔ 𝑡𝑛))
510374, 278, 509syl2anr 595 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 ∈ (ℤ𝑡) ↔ 𝑡𝑛))
511510biimpar 476 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → 𝑛 ∈ (ℤ𝑡))
512511adantlll 716 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → 𝑛 ∈ (ℤ𝑡))
513504, 326, 453syl2an 594 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
514 elfzelz 13541 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℤ)
515514adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℤ)
516 0red 11254 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ∈ ℝ)
517 peano2nn 12262 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℕ)
518517nnred 12265 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℝ)
519518adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ∈ ℝ)
520514zred 12704 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℝ)
521520adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℝ)
522517nngt0d 12299 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 ∈ ℕ → 0 < (𝑡 + 1))
523522adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < (𝑡 + 1))
524 elfzle1 13544 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 ∈ ((𝑡 + 1)...𝑛) → (𝑡 + 1) ≤ 𝑚)
525524adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ≤ 𝑚)
526516, 519, 521, 523, 525ltletrd 11411 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < 𝑚)
527515, 526, 398sylanbrc 581 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
528527adantlr 713 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑡 ∈ ℕ ∧ 𝑡𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
529528adantlll 716 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
530170ffvelcdmda 7093 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓𝑚) ∈ (ℂ × ℂ))
531 ffvelcdm 7090 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓𝑚) ∈ (ℂ × ℂ)) → ( − ‘(𝑓𝑚)) ∈ ℂ)
532137, 530, 531sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → ( − ‘(𝑓𝑚)) ∈ ℂ)
533532absge0d 15435 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → 0 ≤ (abs‘( − ‘(𝑓𝑚))))
534 fvco3 6996 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓𝑚) ∈ (ℂ × ℂ)) → ((abs ∘ − )‘(𝑓𝑚)) = (abs‘( − ‘(𝑓𝑚))))
535137, 530, 534sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → ((abs ∘ − )‘(𝑓𝑚)) = (abs‘( − ‘(𝑓𝑚))))
536505, 535eqtrd 2765 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = (abs‘( − ‘(𝑓𝑚))))
537533, 536breqtrrd 5177 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝑓)‘𝑚))
538537ad5ant15 757 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝑓)‘𝑚))
539529, 538syldan 589 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ≤ (((abs ∘ − ) ∘ 𝑓)‘𝑚))
540471, 512, 513, 539sermono 14040 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
541508, 540eqbrtrd 5171 . . . . . . . . . . . . . . . . . . 19 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
542402ad2antlr 725 . . . . . . . . . . . . . . . . . . 19 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑛 ∈ ℝ)
543478adantl 480 . . . . . . . . . . . . . . . . . . 19 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℝ)
544468, 541, 542, 543ltlecasei 11359 . . . . . . . . . . . . . . . . . 18 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
545544ralrimiva 3135 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∀𝑡 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
546 breq1 5152 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) → (𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
547546ralrn 7097 . . . . . . . . . . . . . . . . . . 19 (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))) Fn ℕ → (∀𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
548355, 547syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (∀𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
549548adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (∀𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
550545, 549mpbird 256 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∀𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
551550r19.21bi 3238 . . . . . . . . . . . . . . 15 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))) → 𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
552361, 362, 551lensymd 11402 . . . . . . . . . . . . . 14 (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))) → ¬ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) < 𝑚)
553311, 322, 358, 552supmax 9497 . . . . . . . . . . . . 13 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))), ℝ*, < ) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
55452, 553sylan 578 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))), ℝ*, < ) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
555255, 309, 5543eqtr3rd 2774 . . . . . . . . . . 11 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) = (vol*‘( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∪ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩))))
556 elfznn 13570 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (1...𝑛) → 𝑧 ∈ ℕ)
557164, 65sselid 3974 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓𝑧) ∈ (ℝ × ℝ))
558 1st2nd2 8033 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑧) ∈ (ℝ × ℝ) → (𝑓𝑧) = ⟨(1st ‘(𝑓𝑧)), (2nd ‘(𝑓𝑧))⟩)
559558fveq2d 6900 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓𝑧)) = ([,]‘⟨(1st ‘(𝑓𝑧)), (2nd ‘(𝑓𝑧))⟩))
560 df-ov 7422 . . . . . . . . . . . . . . . . . . 19 ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) = ([,]‘⟨(1st ‘(𝑓𝑧)), (2nd ‘(𝑓𝑧))⟩)
561559, 560eqtr4di 2783 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓𝑧)) = ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))))
562 xp1st 8026 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑧) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑧)) ∈ ℝ)
563 xp2nd 8027 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑧) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑧)) ∈ ℝ)
564 iccssre 13446 . . . . . . . . . . . . . . . . . . 19 (((1st ‘(𝑓𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓𝑧)) ∈ ℝ) → ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) ⊆ ℝ)
565562, 563, 564syl2anc 582 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) ⊆ ℝ)
566561, 565eqsstrd 4015 . . . . . . . . . . . . . . . . 17 ((𝑓𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓𝑧)) ⊆ ℝ)
567557, 566syl 17 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓𝑧)) ⊆ ℝ)
56852, 556, 567syl2an 594 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓𝑧)) ⊆ ℝ)
569568ralrimiva 3135 . . . . . . . . . . . . . 14 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ ℝ)
570 iunss 5049 . . . . . . . . . . . . . 14 ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ ℝ ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ ℝ)
571569, 570sylibr 233 . . . . . . . . . . . . 13 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ ℝ)
572571adantr 479 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ ℝ)
573 uzid 12875 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℤ → (𝑛 + 1) ∈ (ℤ‘(𝑛 + 1)))
574 ne0i 4334 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ (ℤ‘(𝑛 + 1)) → (ℤ‘(𝑛 + 1)) ≠ ∅)
575 iunconst 5006 . . . . . . . . . . . . . . . 16 ((ℤ‘(𝑛 + 1)) ≠ ∅ → 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩) = ([,]‘⟨0, 0⟩))
576373, 573, 574, 5754syl 19 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩) = ([,]‘⟨0, 0⟩))
577 iccid 13409 . . . . . . . . . . . . . . . . 17 (0 ∈ ℝ* → (0[,]0) = {0})
578259, 577ax-mp 5 . . . . . . . . . . . . . . . 16 (0[,]0) = {0}
579 df-ov 7422 . . . . . . . . . . . . . . . 16 (0[,]0) = ([,]‘⟨0, 0⟩)
580578, 579eqtr3i 2755 . . . . . . . . . . . . . . 15 {0} = ([,]‘⟨0, 0⟩)
581576, 580eqtr4di 2783 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩) = {0})
582 snssi 4813 . . . . . . . . . . . . . . 15 (0 ∈ ℝ → {0} ⊆ ℝ)
583199, 582ax-mp 5 . . . . . . . . . . . . . 14 {0} ⊆ ℝ
584581, 583eqsstrdi 4031 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩) ⊆ ℝ)
585584adantl 480 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩) ⊆ ℝ)
586581fveq2d 6900 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (vol*‘ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = (vol*‘{0}))
587586adantl 480 . . . . . . . . . . . . 13 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = (vol*‘{0}))
588 ovolsn 25485 . . . . . . . . . . . . . 14 (0 ∈ ℝ → (vol*‘{0}) = 0)
589199, 588ax-mp 5 . . . . . . . . . . . . 13 (vol*‘{0}) = 0
590587, 589eqtrdi 2781 . . . . . . . . . . . 12 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = 0)
591 ovolunnul 25490 . . . . . . . . . . . 12 (( 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))))
592572, 585, 590, 591syl3anc 1368 . . . . . . . . . . 11 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∪ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩))) = (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))
593555, 592eqtrd 2765 . . . . . . . . . 10 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) = (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))
594593breq2d 5161 . . . . . . . . 9 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)))))
595594biimpd 228 . . . . . . . 8 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)))))
596595reximdva 3157 . . . . . . 7 (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)))))
597596adantl 480 . . . . . 6 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)))))
598194, 597mpd 15 . . . . 5 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))
599 fzfi 13978 . . . . . . . . . 10 (1...𝑛) ∈ Fin
600 icccld 24744 . . . . . . . . . . . . . . 15 (((1st ‘(𝑓𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓𝑧)) ∈ ℝ) → ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
601562, 563, 600syl2anc 582 . . . . . . . . . . . . . 14 ((𝑓𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
602561, 601eqeltrd 2825 . . . . . . . . . . . . 13 ((𝑓𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
603557, 602syl 17 . . . . . . . . . . . 12 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
604556, 603sylan2 591 . . . . . . . . . . 11 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
605604ralrimiva 3135 . . . . . . . . . 10 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
606 uniretop 24740 . . . . . . . . . . 11 ℝ = (topGen‘ran (,))
607606iuncld 23010 . . . . . . . . . 10 (((topGen‘ran (,)) ∈ Top ∧ (1...𝑛) ∈ Fin ∧ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,)))) → 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
6081, 599, 605, 607mp3an12i 1461 . . . . . . . . 9 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
609608adantr 479 . . . . . . . 8 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))) → 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
610 fveq2 6896 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑓𝑧) → ([,]‘𝑏) = ([,]‘(𝑓𝑧)))
611610sseq1d 4008 . . . . . . . . . . . . . . 15 (𝑏 = (𝑓𝑧) → (([,]‘𝑏) ⊆ 𝐴 ↔ ([,]‘(𝑓𝑧)) ⊆ 𝐴))
612611elrab 3679 . . . . . . . . . . . . . 14 ((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))
613612simprbi 495 . . . . . . . . . . . . 13 ((𝑓𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} → ([,]‘(𝑓𝑧)) ⊆ 𝐴)
61465, 73, 6133syl 18 . . . . . . . . . . . 12 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓𝑧)) ⊆ 𝐴)
615556, 614sylan2 591 . . . . . . . . . . 11 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓𝑧)) ⊆ 𝐴)
616615ralrimiva 3135 . . . . . . . . . 10 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴)
617 iunss 5049 . . . . . . . . . 10 ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴 ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴)
618616, 617sylibr 233 . . . . . . . . 9 (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴)
619618adantr 479 . . . . . . . 8 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))) → 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴)
620 simprr 771 . . . . . . . 8 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))) → 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))
621 sseq1 4002 . . . . . . . . . 10 (𝑠 = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) → (𝑠𝐴 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴))
622 fveq2 6896 . . . . . . . . . . 11 (𝑠 = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) → (vol*‘𝑠) = (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))
623622breq2d 5161 . . . . . . . . . 10 (𝑠 = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)))))
624621, 623anbi12d 630 . . . . . . . . 9 (𝑠 = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) → ((𝑠𝐴𝑀 < (vol*‘𝑠)) ↔ ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))))
625624rspcev 3606 . . . . . . . 8 (( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ∈ (Clsd‘(topGen‘ran (,))) ∧ ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
626609, 619, 620, 625syl12anc 835 . . . . . . 7 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
62752, 626sylan 578 . . . . . 6 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
628627adantll 712 . . . . 5 ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
629598, 628rexlimddv 3150 . . . 4 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
630629adantlr 713 . . 3 ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
63117, 630exlimddv 1930 . 2 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
63215, 631pm2.61dane 3018 1 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3o 1083  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2929  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cun 3942  cin 3943  wss 3944  c0 4322  ifcif 4530  𝒫 cpw 4604  {csn 4630  cop 4636   cuni 4909   ciun 4997  Disj wdisj 5114   class class class wbr 5149  cmpt 5232   Or wor 5589   × cxp 5676  ran crn 5679  cima 5681  ccom 5682   Fn wfn 6544  wf 6545  1-1wf1 6546  ontowfo 6547  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  cmpo 7421  1st c1st 7992  2nd c2nd 7993  Fincfn 8964  supcsup 9470  cc 11143  cr 11144  0cc0 11145  1c1 11146   + caddc 11148  *cxr 11284   < clt 11285  cle 11286  cmin 11481   / cdiv 11908  cn 12250  2c2 12305  0cn0 12510  cz 12596  cuz 12860  (,)cioo 13364  [,]cicc 13367  ...cfz 13524  ..^cfzo 13667  seqcseq 14007  cexp 14067  abscabs 15225  topGenctg 17438  Topctop 22856  Clsdccld 22981  vol*covol 25452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9671  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222  ax-pre-sup 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-omul 8492  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9441  df-sup 9472  df-inf 9473  df-oi 9540  df-dju 9931  df-card 9969  df-acn 9972  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-q 12971  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-ioo 13368  df-ico 13370  df-icc 13371  df-fz 13525  df-fzo 13668  df-fl 13798  df-seq 14008  df-exp 14068  df-hash 14334  df-cj 15090  df-re 15091  df-im 15092  df-sqrt 15226  df-abs 15227  df-clim 15476  df-rlim 15477  df-sum 15677  df-rest 17423  df-topgen 17444  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-top 22857  df-topon 22874  df-bases 22910  df-cld 22984  df-cmp 23352  df-conn 23377  df-ovol 25454  df-vol 25455
This theorem is referenced by:  mblfinlem4  37284  ismblfin  37285
  Copyright terms: Public domain W3C validator