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Theorem mblfinlem2 34319
Description: Lemma for ismblfin 34322, effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different defintion 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 23063 . . . 4 (topGen‘ran (,)) ∈ Top
2 0cld 21340 . . . 4 ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
31, 2ax-mp 5 . . 3 ∅ ∈ (Clsd‘(topGen‘ran (,)))
4 simpl3 1173 . . . . 5 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘𝐴))
5 fveq2 6493 . . . . . 6 (𝐴 = ∅ → (vol*‘𝐴) = (vol*‘∅))
65adantl 474 . . . . 5 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (vol*‘𝐴) = (vol*‘∅))
74, 6breqtrd 4949 . . . 4 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘∅))
8 0ss 4230 . . . 4 ∅ ⊆ 𝐴
97, 8jctil 512 . . 3 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (∅ ⊆ 𝐴𝑀 < (vol*‘∅)))
10 sseq1 3878 . . . . 5 (𝑠 = ∅ → (𝑠𝐴 ↔ ∅ ⊆ 𝐴))
11 fveq2 6493 . . . . . 6 (𝑠 = ∅ → (vol*‘𝑠) = (vol*‘∅))
1211breq2d 4935 . . . . 5 (𝑠 = ∅ → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∅)))
1310, 12anbi12d 621 . . . 4 (𝑠 = ∅ → ((𝑠𝐴𝑀 < (vol*‘𝑠)) ↔ (∅ ⊆ 𝐴𝑀 < (vol*‘∅))))
1413rspcev 3529 . . 3 ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝐴𝑀 < (vol*‘∅))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
153, 9, 14sylancr 578 . 2 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
16 mblfinlem1 34318 . . . 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 1165 . . 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 1173 . . . . . . . . 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 6445 . . . . . . . . . . . . . . 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 5939 . . . . . . . . . . . . . . . . 17 ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
21 forn 6416 . . . . . . . . . . . . . . . . . 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 5764 . . . . . . . . . . . . . . . . 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, 22syl5eq 2820 . . . . . . . . . . . . . . . 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 4716 . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 6977 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦)))
28 oveq1 6977 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1))
2928oveq1d 6985 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦)))
3027, 29opeq12d 4679 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑢 → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩)
31 oveq2 6978 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣))
3231oveq2d 6986 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣)))
3331oveq2d 6986 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣)))
3432, 33opeq12d 4679 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 → ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))⟩)
3530, 34cbvmpov 7059 . . . . . . . . . . . . . . 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 6493 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧))
3736sseq1d 3884 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐)))
38 eqeq1 2776 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑧 → (𝑎 = 𝑐𝑧 = 𝑐))
3937, 38imbi12d 337 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
4039ralbidv 3141 . . . . . . . . . . . . . . . 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 3406 . . . . . . . . . . . . . . 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 3942 . . . . . . . . . . . . . . . 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 23893 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 2811 . . . . . . . . . . . 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 34317 . . . . . . . . . . . . . 14 (𝐴 ∈ (topGen‘ran (,)) → ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
48473ad2ant1 1113 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 2808 . . . . . . . . . . 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 6497 . . . . . . . . . 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 6438 . . . . . . . . . . . . 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 3942 . . . . . . . . . . . . . 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 23885 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
55 frn 6344 . . . . . . . . . . . . . . . 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 3863 . . . . . . . . . . . . . 14 {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ( ≤ ∩ (ℝ × ℝ))
5853, 57sstri 3863 . . . . . . . . . . . . 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 6351 . . . . . . . . . . . . 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 577 . . . . . . . . . . . 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 3863 . . . . . . . . . . . . . . . . . . . 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 ffvelrn 6668 . . . . . . . . . . . . . . . . . . . 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, 62sseldi 3852 . . . . . . . . . . . . . . . . . . 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 704 . . . . . . . . . . . . . . . . . 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 ffvelrn 6668 . . . . . . . . . . . . . . . . . . . 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, 65sseldi 3852 . . . . . . . . . . . . . . . . . . 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 703 . . . . . . . . . . . . . . . . . 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 23890 . . . . . . . . . . . . . . . . . 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 576 . . . . . . . . . . . . . . . . 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 572 . . . . . . . . . . . . . . . 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 1069 . . . . . . . . . . . . . . . 16 ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚)) ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ ((([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) ∨ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))) ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
7270, 71sylib 210 . . . . . . . . . . . . . . 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 3584 . . . . . . . . . . . . . . . . . . . . . 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 6493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = (𝑓𝑚) → ([,]‘𝑎) = ([,]‘(𝑓𝑚)))
7574sseq1d 3884 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = (𝑓𝑚) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓𝑚)) ⊆ ([,]‘𝑐)))
76 eqeq1 2776 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = (𝑓𝑚) → (𝑎 = 𝑐 ↔ (𝑓𝑚) = 𝑐))
7775, 76imbi12d 337 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = (𝑓𝑚) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓𝑚)) ⊆ ([,]‘𝑐) → (𝑓𝑚) = 𝑐)))
7877ralbidv 3141 . . . . . . . . . . . . . . . . . . . . . . . 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 3589 . . . . . . . . . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . . . . . . . . . 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 6493 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (𝑓𝑧) → ([,]‘𝑐) = ([,]‘(𝑓𝑧)))
8281sseq2d 3885 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑓𝑧) → (([,]‘(𝑓𝑚)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧))))
83 eqeq2 2783 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑓𝑧) → ((𝑓𝑚) = 𝑐 ↔ (𝑓𝑚) = (𝑓𝑧)))
8482, 83imbi12d 337 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = (𝑓𝑧) → ((([,]‘(𝑓𝑚)) ⊆ ([,]‘𝑐) → (𝑓𝑚) = 𝑐) ↔ (([,]‘(𝑓𝑚)) ⊆ ([,]‘(𝑓𝑧)) → (𝑓𝑚) = (𝑓𝑧))))
8584rspcva 3527 . . . . . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . . . . . . 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 3584 . . . . . . . . . . . . . . . . . . . . . . 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 6493 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = (𝑓𝑧) → ([,]‘𝑎) = ([,]‘(𝑓𝑧)))
8988sseq1d 3884 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = (𝑓𝑧) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓𝑧)) ⊆ ([,]‘𝑐)))
90 eqeq1 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = (𝑓𝑧) → (𝑎 = 𝑐 ↔ (𝑓𝑧) = 𝑐))
9189, 90imbi12d 337 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = (𝑓𝑧) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓𝑧)) ⊆ ([,]‘𝑐) → (𝑓𝑧) = 𝑐)))
9291ralbidv 3141 . . . . . . . . . . . . . . . . . . . . . . . . 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 3589 . . . . . . . . . . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . . . . . . . . . . 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 6493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = (𝑓𝑚) → ([,]‘𝑐) = ([,]‘(𝑓𝑚)))
9695sseq2d 3885 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (𝑓𝑚) → (([,]‘(𝑓𝑧)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚))))
97 eqeq2 2783 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = (𝑓𝑚) → ((𝑓𝑧) = 𝑐 ↔ (𝑓𝑧) = (𝑓𝑚)))
9896, 97imbi12d 337 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = (𝑓𝑚) → ((([,]‘(𝑓𝑧)) ⊆ ([,]‘𝑐) → (𝑓𝑧) = 𝑐) ↔ (([,]‘(𝑓𝑧)) ⊆ ([,]‘(𝑓𝑚)) → (𝑓𝑧) = (𝑓𝑚))))
9998rspcva 3527 . . . . . . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑧) = (𝑓𝑚) ↔ (𝑓𝑚) = (𝑓𝑧))
102100, 101syl6ib 243 . . . . . . . . . . . . . . . . . . . . 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 845 . . . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . . . . . . 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 572 . . . . . . . . . . . . . . . . 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 6437 . . . . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . . . . 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 572 . . . . . . . . . . . . . . . . 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 948 . . . . . . . . . . . . . . 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 3135 . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 → (𝑚 = 𝑝𝑧 = 𝑝))
115 2fveq3 6498 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑧 → ((,)‘(𝑓𝑚)) = ((,)‘(𝑓𝑧)))
116115ineq1d 4070 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑧 → (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))))
117116eqeq1d 2774 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 → ((((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅ ↔ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅))
118114, 117orbi12d 902 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑧 → ((𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅) ↔ (𝑧 = 𝑝 ∨ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅)))
119118ralbidv 3141 . . . . . . . . . . . . . . 15 (𝑚 = 𝑧 → (∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅)))
120119cbvralv 3377 . . . . . . . . . . . . . 14 (∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅))
121 eqeq2 2783 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑝 → (𝑚 = 𝑧𝑚 = 𝑝))
122 2fveq3 6498 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑝 → ((,)‘(𝑓𝑧)) = ((,)‘(𝑓𝑝)))
123122ineq2d 4071 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑝 → (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))))
124123eqeq1d 2774 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑝 → ((((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅ ↔ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅))
125121, 124orbi12d 902 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑝 → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅)))
126125cbvralv 3377 . . . . . . . . . . . . . . 15 (∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅))
127126ralbii 3109 . . . . . . . . . . . . . 14 (∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ↔ ∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑝))) = ∅))
128122disjor 4905 . . . . . . . . . . . . . 14 (Disj 𝑧 ∈ ℕ ((,)‘(𝑓𝑧)) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓𝑧)) ∩ ((,)‘(𝑓𝑝))) = ∅))
129120, 127, 1283bitr4ri 296 . . . . . . . . . . . . 13 (Disj 𝑧 ∈ ℕ ((,)‘(𝑓𝑧)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅))
130113, 129sylibr 226 . . . . . . . . . . . 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 2772 . . . . . . . . . . . 12 seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
13260, 130, 131uniiccvol 23874 . . . . . . . . . . 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 474 . . . . . . . . . 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 2810 . . . . . . . . 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 4949 . . . . . . . 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 14548 . . . . . . . . . . . 12 abs:ℂ⟶ℝ
137 subf 10680 . . . . . . . . . . . 12 − :(ℂ × ℂ)⟶ℂ
138 fco 6355 . . . . . . . . . . . 12 ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
139136, 137, 138mp2an 679 . . . . . . . . . . 11 (abs ∘ − ):(ℂ × ℂ)⟶ℝ
140 zre 11790 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
141 2re 11507 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
142 reexpcl 13254 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ ℝ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℝ)
143141, 142mpan 677 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℝ)
144 2cn 11508 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℂ
145 2ne0 11544 . . . . . . . . . . . . . . . . . . . . 21 2 ≠ 0
146 nn0z 11811 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℕ0𝑦 ∈ ℤ)
147 expne0i 13269 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0)
148144, 145, 146, 147mp3an12i 1444 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → (2↑𝑦) ≠ 0)
149143, 148jca 504 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0))
150 redivcl 11152 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → (𝑥 / (2↑𝑦)) ∈ ℝ)
151 peano2re 10605 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
152 redivcl 11152 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
153151, 152syl3an1 1143 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
154150, 153opelxpd 5438 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
1551543expb 1100 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0)) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
156140, 149, 155syl2an 586 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
157156rgen2 3147 . . . . . . . . . . . . . . . . 17 𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ)
158 eqid 2772 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
159158fmpo 7567 . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ)
161 frn 6344 . . . . . . . . . . . . . . . 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 3863 . . . . . . . . . . . . . 14 {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ × ℝ)
16453, 163sstri 3863 . . . . . . . . . . . . 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 10384 . . . . . . . . . . . . . 14 ℝ ⊆ ℂ
166 xpss12 5415 . . . . . . . . . . . . . 14 ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
167165, 165, 166mp2an 679 . . . . . . . . . . . . 13 (ℝ × ℝ) ⊆ (ℂ × ℂ)
168164, 167sstri 3863 . . . . . . . . . . . 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 6351 . . . . . . . . . . . 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 678 . . . . . . . . . . 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 6355 . . . . . . . . . . 11 (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ)
172139, 170, 171sylancr 578 . . . . . . . . . 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 12088 . . . . . . . . . . 11 ℕ = (ℤ‘1)
174 1z 11818 . . . . . . . . . . . 12 1 ∈ ℤ
175174a1i 11 . . . . . . . . . . 11 (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → 1 ∈ ℤ)
176 ffvelrn 6668 . . . . . . . . . . 11 ((((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑛) ∈ ℝ)
177173, 175, 176serfre 13207 . . . . . . . . . 10 (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ)
178 frn 6344 . . . . . . . . . . 11 (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ)
179 ressxr 10476 . . . . . . . . . . 11 ℝ ⊆ ℝ*
180178, 179syl6ss 3866 . . . . . . . . . 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 10478 . . . . . . . . . 10 (𝑀 ∈ ℝ → 𝑀 ∈ ℝ*)
1831823ad2ant2 1114 . . . . . . . . 9 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → 𝑀 ∈ ℝ*)
184 supxrlub 12527 . . . . . . . . 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 587 . . . . . . . 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 224 . . . . . . 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 13189 . . . . . . . . . 10 (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ‘1))
188174, 187ax-mp 5 . . . . . . . . 9 seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ‘1)
189173fneq2i 6278 . . . . . . . . 9 (seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ‘1))
190188, 189mpbir 223 . . . . . . . 8 seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ
191 breq2 4927 . . . . . . . . 9 (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → (𝑀 < 𝑧𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
192191rexrn 6672 . . . . . . . 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 210 . . . . . 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))
19560ffvelrnda 6670 . . . . . . . . . . . . . . . 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 11541 . . . . . . . . . . . . . . . . . 18 0 ≤ 0
197 df-br 4924 . . . . . . . . . . . . . . . . . 18 (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
198196, 197mpbi 222 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ ≤
199 0re 10433 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
200 opelxpi 5437 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
201199, 199, 200mp2an 679 . . . . . . . . . . . . . . . . 17 ⟨0, 0⟩ ∈ (ℝ × ℝ)
202 elin 4053 . . . . . . . . . . . . . . . . 17 (⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨0, 0⟩ ∈ ≤ ∧ ⟨0, 0⟩ ∈ (ℝ × ℝ)))
203198, 201, 202mpbir2an 698 . . . . . . . . . . . . . . . 16 ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
204 ifcl 4388 . . . . . . . . . . . . . . . 16 (((𝑓𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
205195, 203, 204sylancl 577 . . . . . . . . . . . . . . 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 6696 . . . . . . . . . . . . . 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 6973 . . . . . . . . . . . . . . . . . . . . . 22 (0(,)0) = ((,)‘⟨0, 0⟩)
208 iooid 12575 . . . . . . . . . . . . . . . . . . . . . 22 (0(,)0) = ∅
209207, 208eqtr3i 2798 . . . . . . . . . . . . . . . . . . . . 21 ((,)‘⟨0, 0⟩) = ∅
210209ineq1i 4067 . . . . . . . . . . . . . . . . . . . 20 (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = (∅ ∩ ((,)‘(𝑓𝑧)))
211 0in 4227 . . . . . . . . . . . . . . . . . . . 20 (∅ ∩ ((,)‘(𝑓𝑧))) = ∅
212210, 211eqtri 2796 . . . . . . . . . . . . . . . . . . 19 (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅
213212olci 852 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅)
214 ineq1 4064 . . . . . . . . . . . . . . . . . . . . 21 (((,)‘(𝑓𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))))
215214eqeq1d 2774 . . . . . . . . . . . . . . . . . . . 20 (((,)‘(𝑓𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → ((((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅))
216215orbi2d 899 . . . . . . . . . . . . . . . . . . 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 4064 . . . . . . . . . . . . . . . . . . . . 21 (((,)‘⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))))
218217eqeq1d 2774 . . . . . . . . . . . . . . . . . . . 20 (((,)‘⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) → ((((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅))
219218orbi2d 899 . . . . . . . . . . . . . . . . . . 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 4382 . . . . . . . . . . . . . . . . . 18 (((𝑚 = 𝑧 ∨ (((,)‘(𝑓𝑚)) ∩ ((,)‘(𝑓𝑧))) = ∅) ∧ (𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓𝑧))) = ∅)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓𝑧))) = ∅))
221112, 213, 220sylancl 577 . . . . . . . . . . . . . . . . 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 4068 . . . . . . . . . . . . . . . . . . 19 (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅)
223 in0 4226 . . . . . . . . . . . . . . . . . . 19 (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅) = ∅
224222, 223eqtri 2796 . . . . . . . . . . . . . . . . . 18 (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅
225224olci 852 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅)
226 ineq2 4065 . . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . . 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 899 . . . . . . . . . . . . . . . . . 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 4065 . . . . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . . . . 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 899 . . . . . . . . . . . . . . . . . 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 4382 . . . . . . . . . . . . . . . . 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 577 . . . . . . . . . . . . . . . 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 3135 . . . . . . . . . . . . . . 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 4895 . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑚 → (𝑧 ∈ (1...𝑛) ↔ 𝑚 ∈ (1...𝑛)))
237 fveq2 6493 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝑚 → (𝑓𝑧) = (𝑓𝑚))
238236, 237ifbieq1d 4367 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑚 → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩))
239 eqid 2772 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))
240 fvex 6506 . . . . . . . . . . . . . . . . . . . . 21 (𝑓𝑚) ∈ V
241 opex 5206 . . . . . . . . . . . . . . . . . . . . 21 ⟨0, 0⟩ ∈ V
242240, 241ifex 4392 . . . . . . . . . . . . . . . . . . . 20 if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩) ∈ V
243238, 239, 242fvmpt 6589 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩))
244243fveq2d 6497 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ ℕ → ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)) = ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩)))
245 fvif 6509 . . . . . . . . . . . . . . . . . 18 ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩))
246244, 245syl6eq 2824 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ → ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)))
247235, 246mprg 3096 . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑧 → (𝑚 ∈ (1...𝑛) ↔ 𝑧 ∈ (1...𝑛)))
249248, 115ifbieq1d 4367 . . . . . . . . . . . . . . . . 17 (𝑚 = 𝑧 → if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓𝑚)), ((,)‘⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓𝑧)), ((,)‘⟨0, 0⟩)))
250249disjor 4905 . . . . . . . . . . . . . . . 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 267 . . . . . . . . . . . . . . 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 226 . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . 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 23874 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 10477 . . . . . . . . . . . . . . . . . . . . 21 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
257164, 256sstri 3863 . . . . . . . . . . . . . . . . . . . 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, 65sseldi 3852 . . . . . . . . . . . . . . . . . . 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 10479 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℝ*
260 opelxpi 5437 . . . . . . . . . . . . . . . . . . . 20 ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ⟨0, 0⟩ ∈ (ℝ* × ℝ*))
261259, 259, 260mp2an 679 . . . . . . . . . . . . . . . . . . 19 ⟨0, 0⟩ ∈ (ℝ* × ℝ*)
262 ifcl 4388 . . . . . . . . . . . . . . . . . . 19 (((𝑓𝑧) ∈ (ℝ* × ℝ*) ∧ ⟨0, 0⟩ ∈ (ℝ* × ℝ*)) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ (ℝ* × ℝ*))
263258, 261, 262sylancl 577 . . . . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . . . . 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 12645 . . . . . . . . . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . . . . . 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 6493 . . . . . . . . . . . . . . . . . 18 (𝑚 = if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) → ([,]‘𝑚) = ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))
269263, 264, 267, 268fmptco 6708 . . . . . . . . . . . . . . . . 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 5644 . . . . . . . . . . . . . . 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 4716 . . . . . . . . . . . . . 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 11445 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
274273, 173syl6eleq 2870 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ (ℤ‘1))
275 fzouzsplit 12880 . . . . . . . . . . . . . . . . . . . 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, 276syl5eq 2820 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ℕ = ((1..^(𝑛 + 1)) ∪ (ℤ‘(𝑛 + 1))))
278 nnz 11810 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
279 fzval3 12914 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℤ → (1...𝑛) = (1..^(𝑛 + 1)))
280278, 279syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → (1...𝑛) = (1..^(𝑛 + 1)))
281280uneq1d 4023 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → ((1...𝑛) ∪ (ℤ‘(𝑛 + 1))) = ((1..^(𝑛 + 1)) ∪ (ℤ‘(𝑛 + 1))))
282277, 281eqtr4d 2811 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → ℕ = ((1...𝑛) ∪ (ℤ‘(𝑛 + 1))))
283 fvif 6509 . . . . . . . . . . . . . . . . . 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 4813 . . . . . . . . . . . . . . . 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 6506 . . . . . . . . . . . . . . . . 17 ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) ∈ V
287286dfiun3 5672 . . . . . . . . . . . . . . . 16 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))
288 iunxun 4876 . . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . . . 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 4350 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘(𝑓𝑧)))
291290iuneq2i 4806 . . . . . . . . . . . . . . . . 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 12799 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (ℤ‘(𝑛 + 1)) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
294293adantl 474 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
295 nncn 11440 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
296 ax-1cn 10385 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℂ
297 pncan 10684 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
298295, 296, 297sylancl 577 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛)
299298oveq2d 6986 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ℕ → (1...((𝑛 + 1) − 1)) = (1...𝑛))
300299eleq2d 2845 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℕ → (𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ 𝑧 ∈ (1...𝑛)))
301300notbid 310 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
302301adantr 473 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ‘(𝑛 + 1))) → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
303294, 302mpbid 224 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...𝑛))
304303iffalsed 4355 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ‘(𝑛 + 1))) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘⟨0, 0⟩))
305304iuneq2dv 4809 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → 𝑧 ∈ (ℤ‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓𝑧)), ([,]‘⟨0, 0⟩)) = 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩))
306292, 305uneq12d 4025 . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . 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 6497 . . . . . . . . . . . 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 12344 . . . . . . . . . . . . . . 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 12089 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
313312biimpi 208 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ‘1))
314313adantl 474 . . . . . . . . . . . . . . . 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 12745 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ (1...𝑛) → 𝑢 ∈ ℕ)
316172ffvelrnda 6670 . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . 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 702 . . . . . . . . . . . . . . . 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 10410 . . . . . . . . . . . . . . . . 17 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ)
320319adantl 474 . . . . . . . . . . . . . . . 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 13198 . . . . . . . . . . . . . . 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 10482 . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (1...𝑛) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)))
324 iftrue 4350 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (1...𝑛) → if(𝑚 ∈ (1...𝑛), (𝑓𝑚), ⟨0, 0⟩) = (𝑓𝑚))
325238, 324sylan9eqr 2830 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 ∈ (1...𝑛) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = (𝑓𝑚))
326 elfznn 12745 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (1...𝑛) → 𝑚 ∈ ℕ)
327240a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ (1...𝑛) → (𝑓𝑚) ∈ V)
328323, 325, 326, 327fvmptd 6595 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ (1...𝑛) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓𝑚))
329328adantl 474 . . . . . . . . . . . . . . . . . . 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 6497 . . . . . . . . . . . . . . . . . 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 6506 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝑧) ∈ V
332331, 241ifex 4392 . . . . . . . . . . . . . . . . . . . . 21 if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ V
333332, 239fnmpti 6315 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩)) Fn ℕ
334 fvco2 6580 . . . . . . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . 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 6338 . . . . . . . . . . . . . . . . . . 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 6580 . . . . . . . . . . . . . . . . . . 19 ((𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓𝑚)))
339337, 326, 338syl2an 586 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓𝑚)))
340330, 336, 3393eqtr4d 2818 . . . . . . . . . . . . . . . . 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 702 . . . . . . . . . . . . . . . 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 13202 . . . . . . . . . . . . . . 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, 65sseldi 3852 . . . . . . . . . . . . . . . . . . . . . 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 10423 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
346 opelxpi 5437 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨0, 0⟩ ∈ (ℂ × ℂ))
347345, 345, 346mp2an 679 . . . . . . . . . . . . . . . . . . . . . 22 ⟨0, 0⟩ ∈ (ℂ × ℂ)
348 ifcl 4388 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓𝑧) ∈ (ℂ × ℂ) ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) ∈ (ℂ × ℂ))
349344, 347, 348sylancl 577 . . . . . . . . . . . . . . . . . . . . 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 6696 . . . . . . . . . . . . . . . . . . . 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 6355 . . . . . . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . . . . . . . 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)
353352ffvelrnda 6670 . . . . . . . . . . . . . . . . . 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 13207 . . . . . . . . . . . . . . . . 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 6339 . . . . . . . . . . . . . . . 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 6667 . . . . . . . . . . . . . . . 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 572 . . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . . 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 6345 . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . 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 3854 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 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 10410 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑚 + 𝑢) ∈ ℝ)
364363adantl 474 . . . . . . . . . . . . . . . . . . . . . 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 10417 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 ∈ ℝ → 𝑚 ∈ ℂ)
366 recn 10417 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
367 recn 10417 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ ℝ → 𝑣 ∈ ℂ)
368 addass 10414 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
369365, 366, 367, 368syl3an 1140 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
370369adantl 474 . . . . . . . . . . . . . . . . . . . . . 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 11844 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 < 𝑡 ↔ (𝑛 + 1) ≤ 𝑡))
372371biimpa 469 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑛 + 1) ≤ 𝑡)
373273nnzd 11892 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℤ)
374 nnz 11810 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ ℕ → 𝑡 ∈ ℤ)
375 eluz 12065 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑛 + 1) ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ∈ (ℤ‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
376373, 374, 375syl2an 586 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (ℤ‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
377376adantr 473 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑡 ∈ (ℤ‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
378372, 377mpbird 249 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ‘(𝑛 + 1)))
379378adantlll 705 . . . . . . . . . . . . . . . . . . . . . 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 718 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑛 ∈ (ℤ‘1))
381 simplll 762 . . . . . . . . . . . . . . . . . . . . . . 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 12745 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℕ)
383381, 382, 353syl2an 586 . . . . . . . . . . . . . . . . . . . . . 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 13211 . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . . 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 12717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℤ)
387386adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
388 0red 10435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
389273nnred 11448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
390389ad3antrrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
391386zred 11893 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℝ)
392391adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
393273nngt0d 11482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑛 ∈ ℕ → 0 < (𝑛 + 1))
394393ad3antrrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
395 elfzle1 12719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 ∈ ((𝑛 + 1)...𝑡) → (𝑛 + 1) ≤ 𝑚)
396395adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
397388, 390, 392, 394, 396ltletrd 10592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
398 elnnz 11796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 ∈ ℕ ↔ (𝑚 ∈ ℤ ∧ 0 < 𝑚))
399387, 397, 398sylanbrc 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ)
400333, 399, 334sylancr 578 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
403402adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 ∈ ℝ)
404389adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
405391adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
406402ltp1d 11363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
407406adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < (𝑛 + 1))
408395adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
409403, 404, 405, 407, 408ltletrd 10592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < 𝑚)
410409adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → 𝑛 < 𝑚)
411403, 405ltnled 10579 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 < 𝑚 ↔ ¬ 𝑚𝑛))
412 breq1 4926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑚 = 𝑧 → (𝑚𝑛𝑧𝑛))
413412equcoms 1976 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑧 = 𝑚 → (𝑚𝑛𝑧𝑛))
414413notbid 310 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 = 𝑚 → (¬ 𝑚𝑛 ↔ ¬ 𝑧𝑛))
415411, 414sylan9bb 502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → (𝑛 < 𝑚 ↔ ¬ 𝑧𝑛))
416410, 415mpbid 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧𝑛)
417 elfzle2 12720 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 ∈ (1...𝑛) → 𝑧𝑛)
418416, 417nsyl 138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ∈ (1...𝑛))
419418iffalsed 4355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩) = ⟨0, 0⟩)
420386adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
421 0red 10435 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
422393adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
423421, 404, 405, 422, 408ltletrd 10592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
424420, 423, 398sylanbrc 575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = ⟨0, 0⟩)
427426ad4ant14 739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))‘𝑚) = ⟨0, 0⟩)
428427fveq2d 6497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (( − :(ℂ × ℂ)⟶ℂ ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩)))
431137, 347, 430mp2an 679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩))
432 df-ov 6973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 − 0) = ( − ‘⟨0, 0⟩)
433 0m0e0 11560 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (0 − 0) = 0
434432, 433eqtr3i 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ( − ‘⟨0, 0⟩) = 0
435434fveq2i 6496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (abs‘( − ‘⟨0, 0⟩)) = (abs‘0)
436 abs0 14496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (abs‘0) = 0
437435, 436eqtri 2796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (abs‘( − ‘⟨0, 0⟩)) = 0
438431, 437eqtri 2796 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((abs ∘ − )‘⟨0, 0⟩) = 0
439429, 438syl6eq 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ (ℤ‘(𝑛 + 1)))
441 c0ex 10425 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 ∈ V
442441fvconst2 6787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((ℤ‘(𝑛 + 1)) × {0})‘𝑚) = 0)
445439, 444eqtr4d 2811 . . . . . . . . . . . . . . . . . . . . . . . . . 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 13202 . . . . . . . . . . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (ℤ‘(𝑛 + 1)) = (ℤ‘(𝑛 + 1))
448447ser0 13230 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓𝑧), ⟨0, 0⟩))))‘𝑡) = 0)
451450adantlll 705 . . . . . . . . . . . . . . . . . . . . . . 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 6988 . . . . . . . . . . . . . . . . . . . . . 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))
453172ffvelrnda 6670 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 702 . . . . . . . . . . . . . . . . . . . . . . . . . 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 10410 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑚 + 𝑣) ∈ ℝ)
457456adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . 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 13198 . . . . . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . . . 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 10460 . . . . . . . . . . . . . . . . . . . . . . 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)
461460addid1d 10632 . . . . . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . . . 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 13198 . . . . . . . . . . . . . . . . . . . . 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 10999 . . . . . . . . . . . . . . . . . . . 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 4945 . . . . . . . . . . . . . . . . . . 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 12089 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 ∈ ℕ ↔ 𝑡 ∈ (ℤ‘1))
470469biimpi 208 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 ∈ ℕ → 𝑡 ∈ (ℤ‘1))
471470ad2antlr 714 . . . . . . . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 ∈ (1...𝑡) → 1 ≤ 𝑚)
475474adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 1 ≤ 𝑚)
476382nnred 11448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℝ)
477476adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℝ)
478 nnre 11439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 ∈ ℕ → 𝑡 ∈ ℝ)
479478ad3antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ∈ ℝ)
480402ad3antrrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑛 ∈ ℝ)
481 elfzle2 12720 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 ∈ (1...𝑡) → 𝑚𝑡)
482481adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚𝑡)
483 simplr 756 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡𝑛)
484477, 479, 480, 482, 483letrd 10589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚𝑛)
485 elfzelz 12717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℤ)
486278ad2antrr 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → 𝑛 ∈ ℤ)
487 elfz 12707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑚 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚𝑚𝑛)))
488174, 487mp3an2 1428 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚𝑚𝑛)))
489485, 486, 488syl2anr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚𝑚𝑛)))
490475, 484, 489mpbir2and 700 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛))
491490ad5ant2345 1350 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2860 . . . . . . . . . . . . . . . . . . . . . . . . . 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 4350 . . . . . . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . . . 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 6595 . . . . . . . . . . . . . . . . . . . . . . 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 6497 . . . . . . . . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . . . . 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 6582 . . . . . . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . . . . . . 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 2818 . . . . . . . . . . . . . . . . . . . . 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 13202 . . . . . . . . . . . . . . . . . . . 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 12065 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ𝑡) ↔ 𝑡𝑛))
510374, 278, 509syl2anr 587 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 ∈ (ℤ𝑡) ↔ 𝑡𝑛))
511510biimpar 470 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) → 𝑛 ∈ (ℤ𝑡))
512511adantlll 705 . . . . . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡𝑛) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
514 elfzelz 12717 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℤ)
515514adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℤ)
516 0red 10435 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ∈ ℝ)
517 peano2nn 11445 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℕ)
518517nnred 11448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℝ)
519518adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ∈ ℝ)
520514zred 11893 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℝ)
521520adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℝ)
522517nngt0d 11482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 ∈ ℕ → 0 < (𝑡 + 1))
523522adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < (𝑡 + 1))
524 elfzle1 12719 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 ∈ ((𝑡 + 1)...𝑛) → (𝑡 + 1) ≤ 𝑚)
525524adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ≤ 𝑚)
526516, 519, 521, 523, 525ltletrd 10592 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < 𝑚)
527515, 526, 398sylanbrc 575 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
528527adantlr 702 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑡 ∈ ℕ ∧ 𝑡𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
529528adantlll 705 . . . . . . . . . . . . . . . . . . . . . 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)
530170ffvelrnda 6670 . . . . . . . . . . . . . . . . . . . . . . . . . 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 ffvelrn 6668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓𝑚) ∈ (ℂ × ℂ)) → ( − ‘(𝑓𝑚)) ∈ ℂ)
532137, 530, 531sylancr 578 . . . . . . . . . . . . . . . . . . . . . . . . 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 14655 . . . . . . . . . . . . . . . . . . . . . . . 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 6582 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓𝑚) ∈ (ℂ × ℂ)) → ((abs ∘ − )‘(𝑓𝑚)) = (abs‘( − ‘(𝑓𝑚))))
535137, 530, 534sylancr 578 . . . . . . . . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . . . . . . . . . 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 4951 . . . . . . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . . . . . . . . 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 13210 . . . . . . . . . . . . . . . . . . . 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 4945 . . . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . 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 10540 . . . . . . . . . . . . . . . . . 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 3126 . . . . . . . . . . . . . . . . 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 4926 . . . . . . . . . . . . . . . . . . . 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 6673 . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . . . 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 3152 . . . . . . . . . . . . . . 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 10583 . . . . . . . . . . . . . 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 8718 . . . . . . . . . . . . 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 572 . . . . . . . . . . . 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 2817 . . . . . . . . . . 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 12745 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (1...𝑛) → 𝑧 ∈ ℕ)
557164, 65sseldi 3852 . . . . . . . . . . . . . . . . 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 7533 . . . . . . . . . . . . . . . . . . . 20 ((𝑓𝑧) ∈ (ℝ × ℝ) → (𝑓𝑧) = ⟨(1st ‘(𝑓𝑧)), (2nd ‘(𝑓𝑧))⟩)
559558fveq2d 6497 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓𝑧)) = ([,]‘⟨(1st ‘(𝑓𝑧)), (2nd ‘(𝑓𝑧))⟩))
560 df-ov 6973 . . . . . . . . . . . . . . . . . . 19 ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) = ([,]‘⟨(1st ‘(𝑓𝑧)), (2nd ‘(𝑓𝑧))⟩)
561559, 560syl6eqr 2826 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓𝑧)) = ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))))
562 xp1st 7526 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑧) ∈ (ℝ × ℝ) → (1st ‘(𝑓𝑧)) ∈ ℝ)
563 xp2nd 7527 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑧) ∈ (ℝ × ℝ) → (2nd ‘(𝑓𝑧)) ∈ ℝ)
564 iccssre 12627 . . . . . . . . . . . . . . . . . . 19 (((1st ‘(𝑓𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓𝑧)) ∈ ℝ) → ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) ⊆ ℝ)
565562, 563, 564syl2anc 576 . . . . . . . . . . . . . . . . . 18 ((𝑓𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) ⊆ ℝ)
566561, 565eqsstrd 3891 . . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓𝑧)) ⊆ ℝ)
569568ralrimiva 3126 . . . . . . . . . . . . . 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 4829 . . . . . . . . . . . . . 14 ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ ℝ ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ ℝ)
571569, 570sylibr 226 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 12066 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ ℤ → (𝑛 + 1) ∈ (ℤ‘(𝑛 + 1)))
574 ne0i 4181 . . . . . . . . . . . . . . . 16 ((𝑛 + 1) ∈ (ℤ‘(𝑛 + 1)) → (ℤ‘(𝑛 + 1)) ≠ ∅)
575 iunconst 4795 . . . . . . . . . . . . . . . 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 12592 . . . . . . . . . . . . . . . . 17 (0 ∈ ℝ* → (0[,]0) = {0})
578259, 577ax-mp 5 . . . . . . . . . . . . . . . 16 (0[,]0) = {0}
579 df-ov 6973 . . . . . . . . . . . . . . . 16 (0[,]0) = ([,]‘⟨0, 0⟩)
580578, 579eqtr3i 2798 . . . . . . . . . . . . . . 15 {0} = ([,]‘⟨0, 0⟩)
581576, 580syl6eqr 2826 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩) = {0})
582 snssi 4609 . . . . . . . . . . . . . . 15 (0 ∈ ℝ → {0} ⊆ ℝ)
583199, 582ax-mp 5 . . . . . . . . . . . . . 14 {0} ⊆ ℝ
584581, 583syl6eqss 3907 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩) ⊆ ℝ)
585584adantl 474 . . . . . . . . . . . 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 6497 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (vol*‘ 𝑧 ∈ (ℤ‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = (vol*‘{0}))
587586adantl 474 . . . . . . . . . . . . 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 23789 . . . . . . . . . . . . . 14 (0 ∈ ℝ → (vol*‘{0}) = 0)
589199, 588ax-mp 5 . . . . . . . . . . . . 13 (vol*‘{0}) = 0
590587, 589syl6eq 2824 . . . . . . . . . . . 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 23794 . . . . . . . . . . . 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 1351 . . . . . . . . . . 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 2808 . . . . . . . . . 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 4935 . . . . . . . . 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 221 . . . . . . . 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 3213 . . . . . . 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 474 . . . . . 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 13148 . . . . . . . . . 10 (1...𝑛) ∈ Fin
600 icccld 23068 . . . . . . . . . . . . . . 15 (((1st ‘(𝑓𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓𝑧)) ∈ ℝ) → ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
601562, 563, 600syl2anc 576 . . . . . . . . . . . . . 14 ((𝑓𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓𝑧))[,](2nd ‘(𝑓𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
602561, 601eqeltrd 2860 . . . . . . . . . . . . 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 583 . . . . . . . . . . 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 3126 . . . . . . . . . 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 23064 . . . . . . . . . . 11 ℝ = (topGen‘ran (,))
607606iuncld 21347 . . . . . . . . . 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 1444 . . . . . . . . 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 473 . . . . . . . 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 6493 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑓𝑧) → ([,]‘𝑏) = ([,]‘(𝑓𝑧)))
611610sseq1d 3884 . . . . . . . . . . . . . . 15 (𝑏 = (𝑓𝑧) → (([,]‘𝑏) ⊆ 𝐴 ↔ ([,]‘(𝑓𝑧)) ⊆ 𝐴))
612611elrab 3589 . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . 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 583 . . . . . . . . . . 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 3126 . . . . . . . . . 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 4829 . . . . . . . . . 10 ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴 ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴)
618616, 617sylibr 226 . . . . . . . . 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 473 . . . . . . . 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 760 . . . . . . . 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 3878 . . . . . . . . . 10 (𝑠 = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) → (𝑠𝐴 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴))
622 fveq2 6493 . . . . . . . . . . 11 (𝑠 = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) → (vol*‘𝑠) = (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))
623622breq2d 4935 . . . . . . . . . 10 (𝑠 = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)))))
624621, 623anbi12d 621 . . . . . . . . 9 (𝑠 = 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) → ((𝑠𝐴𝑀 < (vol*‘𝑠)) ↔ ( 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧)) ⊆ 𝐴𝑀 < (vol*‘ 𝑧 ∈ (1...𝑛)([,]‘(𝑓𝑧))))))
625624rspcev 3529 . . . . . . . 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 824 . . . . . . 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 572 . . . . . 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 701 . . . . 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 3230 . . . 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 702 . . 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 1894 . 2 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠𝐴𝑀 < (vol*‘𝑠)))
63215, 631pm2.61dane 3049 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 198  wa 387  wo 833  w3o 1067  w3a 1068   = wceq 1507  wex 1742  wcel 2048  wne 2961  wral 3082  wrex 3083  {crab 3086  Vcvv 3409  cun 3823  cin 3824  wss 3825  c0 4173  ifcif 4344  𝒫 cpw 4416  {csn 4435  cop 4441   cuni 4706   ciun 4786  Disj wdisj 4891   class class class wbr 4923  cmpt 5002   Or wor 5318   × cxp 5398  ran crn 5401  cima 5403  ccom 5404   Fn wfn 6177  wf 6178  1-1wf1 6179  ontowfo 6180  1-1-ontowf1o 6181  cfv 6182  (class class class)co 6970  cmpo 6972  1st c1st 7492  2nd c2nd 7493  Fincfn 8298  supcsup 8691  cc 10325  cr 10326  0cc0 10327  1c1 10328   + caddc 10330  *cxr 10465   < clt 10466  cle 10467  cmin 10662   / cdiv 11090  cn 11431  2c2 11488  0cn0 11700  cz 11786  cuz 12051  (,)cioo 12547  [,]cicc 12550  ...cfz 12701  ..^cfzo 12842  seqcseq 13177  cexp 13237  abscabs 14444  topGenctg 16557  Topctop 21195  Clsdccld 21318  vol*covol 23756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8890  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-disj 4892  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-2o 7898  df-oadd 7901  df-omul 7902  df-er 8081  df-map 8200  df-pm 8201  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-fi 8662  df-sup 8693  df-inf 8694  df-oi 8761  df-dju 9116  df-card 9154  df-acn 9157  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-n0 11701  df-z 11787  df-uz 12052  df-q 12156  df-rp 12198  df-xneg 12317  df-xadd 12318  df-xmul 12319  df-ioo 12551  df-ico 12553  df-icc 12554  df-fz 12702  df-fzo 12843  df-fl 12970  df-seq 13178  df-exp 13238  df-hash 13499  df-cj 14309  df-re 14310  df-im 14311  df-sqrt 14445  df-abs 14446  df-clim 14696  df-rlim 14697  df-sum 14894  df-rest 16542  df-topgen 16563  df-psmet 20229  df-xmet 20230  df-met 20231  df-bl 20232  df-mopn 20233  df-top 21196  df-topon 21213  df-bases 21248  df-cld 21321  df-cmp 21689  df-conn 21714  df-ovol 23758  df-vol 23759
This theorem is referenced by:  mblfinlem4  34321  ismblfin  34322
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