Theorem mblfinlem2 37659

Description: Lemma for ismblfin 37662, effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different definition of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)

Assertion

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

Step	Hyp	Ref	Expression
1		retop 24656	. . . 4 ⊢ (topGen‘ran (,)) ∈ Top
2		0cld 22932	. . . 4 ⊢ ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
3	1, 2	ax-mp 5	. . 3 ⊢ ∅ ∈ (Clsd‘(topGen‘ran (,)))
4		simpl3 1194	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘𝐴))
5		fveq2 6861	. . . . . 6 ⊢ (𝐴 = ∅ → (vol*‘𝐴) = (vol*‘∅))
6	5	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (vol*‘𝐴) = (vol*‘∅))
7	4, 6	breqtrd 5136	. . . 4 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘∅))
8		0ss 4366	. . . 4 ⊢ ∅ ⊆ 𝐴
9	7, 8	jctil 519	. . 3 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅)))
10		sseq1 3975	. . . . 5 ⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
11		fveq2 6861	. . . . . 6 ⊢ (𝑠 = ∅ → (vol*‘𝑠) = (vol*‘∅))
12	11	breq2d 5122	. . . . 5 ⊢ (𝑠 = ∅ → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∅)))
13	10, 12	anbi12d 632	. . . 4 ⊢ (𝑠 = ∅ → ((𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)) ↔ (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅))))
14	13	rspcev 3591	. . 3 ⊢ ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
15	3, 9, 14	sylancr 587	. 2 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
16		mblfinlem1 37658	. . . 4 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
17	16	3ad2antl1 1186	. . 3 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
18		simpl3 1194	. . . . . . . . 9 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑀 < (vol*‘𝐴))
19		f1ofo 6810	. . . . . . . . . . . . . . 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 6229	. . . . . . . . . . . . . . . . 17 ⊢ ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
21		forn 6778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran 𝑓 = {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
22	21	imaeq2d 6034	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] “ ran 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
23	20, 22	eqtrid 2777	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
24	23	unieqd 4887	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran ([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
25	19, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran ([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
26	25	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran ([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
27		oveq1 7397	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦)))
28		oveq1 7397	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1))
29	28	oveq1d 7405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦)))
30	27, 29	opeq12d 4848	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩)
31		oveq2 7398	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣))
32	31	oveq2d 7406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣)))
33	31	oveq2d 7406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣)))
34	32, 33	opeq12d 4848	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))⟩)
35	30, 34	cbvmpov 7487	. . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧))
37	36	sseq1d 3981	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐)))
38		eqeq1 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (𝑎 = 𝑐 ↔ 𝑧 = 𝑐))
39	37, 38	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
40	39	ralbidv 3157	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑧 → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
41	40	cbvrabv 3419	. . . . . . . . . . . . . . 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 4046	. . . . . . . . . . . . . . . 16 ⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
43	42	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
44	35, 41, 43	dyadmbllem 25507	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
45	44	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
46	26, 45	eqtr4d 2768	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran ([,] ∘ 𝑓) = ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}))
47		opnmbllem0 37657	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
48	47	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
49	48	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
50	46, 49	eqtrd 2765	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran ([,] ∘ 𝑓) = 𝐴)
51	50	fveq2d 6865	. . . . . . . . . 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 6803	. . . . . . . . . . . . 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 4046	. . . . . . . . . . . . . 14 ⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}
54	35	dyadf 25499	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
55		frn 6698	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ ( ≤ ∩ (ℝ × ℝ)))
56	54, 55	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ ( ≤ ∩ (ℝ × ℝ))
57	42, 56	sstri 3959	. . . . . . . . . . . . . 14 ⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ( ≤ ∩ (ℝ × ℝ))
58	53, 57	sstri 3959	. . . . . . . . . . . . 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 6707	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ( ≤ ∩ (ℝ × ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
60	52, 58, 59	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
61	53, 42	sstri 3959	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
62		ffvelcdm 7056	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
63	61, 62	sselid 3947	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
64	63	adantrr 717	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
65		ffvelcdm 7056	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
66	61, 65	sselid 3947	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
67	66	adantrl 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
68	35	dyaddisj 25504	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∧ (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
69	64, 67, 68	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
70	52, 69	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
71		df-3or 1087	. . . . . . . . . . . . . . . 16 ⊢ ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
72	70, 71	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
73		elrabi 3657	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴})
74		fveq2 6861	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑚) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑚)))
75	74	sseq1d 3981	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑚) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐)))
76		eqeq1 2734	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑚) → (𝑎 = 𝑐 ↔ (𝑓‘𝑚) = 𝑐))
77	75, 76	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝑓‘𝑚) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)))
78	77	ralbidv 3157	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝑓‘𝑚) → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)))
79	78	elrab 3662	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ ((𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)))
80	79	simprbi 496	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐))
81		fveq2 6861	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑧) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑧)))
82	81	sseq2d 3982	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑧) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧))))
83		eqeq2 2742	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑧) → ((𝑓‘𝑚) = 𝑐 ↔ (𝑓‘𝑚) = (𝑓‘𝑧)))
84	82, 83	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (𝑓‘𝑧) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧))))
85	84	rspcva 3589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧)))
86	73, 80, 85	syl2anr 597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧)))
87		elrabi 3657	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴})
88		fveq2 6861	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = (𝑓‘𝑧) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑧)))
89	88	sseq1d 3981	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑧) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐)))
90		eqeq1 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑧) → (𝑎 = 𝑐 ↔ (𝑓‘𝑧) = 𝑐))
91	89, 90	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑧) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)))
92	91	ralbidv 3157	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝑓‘𝑧) → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)))
93	92	elrab 3662	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)))
94	93	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐))
95		fveq2 6861	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 = (𝑓‘𝑚) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑚)))
96	95	sseq2d 3982	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑚) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))))
97		eqeq2 2742	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑚) → ((𝑓‘𝑧) = 𝑐 ↔ (𝑓‘𝑧) = (𝑓‘𝑚)))
98	96, 97	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑚) → ((([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚))))
99	98	rspcva 3589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚)))
100	87, 94, 99	syl2an 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚)))
101		eqcom 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑧) = (𝑓‘𝑚) ↔ (𝑓‘𝑚) = (𝑓‘𝑧))
102	100, 101	imbitrdi 251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑚) = (𝑓‘𝑧)))
103	86, 102	jaod 859	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧)))
104	62, 65, 103	syl2an 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) ∧ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧)))
105	104	anandis 678	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧)))
106	52, 105	sylan 580	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧)))
107		f1of1 6802	. . . . . . . . . . . . . . . . . 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 7234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ–1-1→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑓‘𝑚) = (𝑓‘𝑧) → 𝑚 = 𝑧))
109	107, 108	sylan 580	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑓‘𝑚) = (𝑓‘𝑧) → 𝑚 = 𝑧))
110	106, 109	syld 47	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → 𝑚 = 𝑧))
111	110	orim1d 967	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) → (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)))
112	72, 111	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
113	112	ralrimivva 3181	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
114		eqeq1 2734	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → (𝑚 = 𝑝 ↔ 𝑧 = 𝑝))
115		2fveq3 6866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → ((,)‘(𝑓‘𝑚)) = ((,)‘(𝑓‘𝑧)))
116	115	ineq1d 4185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))))
117	116	eqeq1d 2732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅ ↔ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
118	114, 117	orbi12d 918	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑧 → ((𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
119	118	ralbidv 3157	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑧 → (∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
120	119	cbvralvw 3216	. . . . . . . . . . . . . 14 ⊢ (∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
121		eqeq2 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑝 → (𝑚 = 𝑧 ↔ 𝑚 = 𝑝))
122		2fveq3 6866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑝 → ((,)‘(𝑓‘𝑧)) = ((,)‘(𝑓‘𝑝)))
123	122	ineq2d 4186	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑝 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))))
124	123	eqeq1d 2732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑝 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
125	121, 124	orbi12d 918	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑝 → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
126	125	cbvralvw 3216	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
127	126	ralbii 3076	. . . . . . . . . . . . . 14 ⊢ (∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
128	122	disjor 5092	. . . . . . . . . . . . . 14 ⊢ (Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
129	120, 127, 128	3bitr4ri 304	. . . . . . . . . . . . 13 ⊢ (Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
130	113, 129	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧)))
131		eqid 2730	. . . . . . . . . . . 12 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
132	60, 130, 131	uniiccvol 25488	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (vol*‘∪ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
133	132	adantl 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘∪ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
134	51, 133	eqtr3d 2767	. . . . . . . . 9 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘𝐴) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
135	18, 134	breqtrd 5136	. . . . . . . 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 15311	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
137		subf 11430	. . . . . . . . . . . 12 ⊢ − :(ℂ × ℂ)⟶ℂ
138		fco 6715	. . . . . . . . . . . 12 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
139	136, 137, 138	mp2an 692	. . . . . . . . . . 11 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
140		zre 12540	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
141		2re 12267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
142		reexpcl 14050	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℝ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℝ)
143	141, 142	mpan 690	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℝ)
144		2cn 12268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℂ
145		2ne0 12297	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
146		nn0z 12561	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ)
147		expne0i 14066	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0)
148	144, 145, 146, 147	mp3an12i 1467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → (2↑𝑦) ≠ 0)
149	143, 148	jca 511	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0))
150		redivcl 11908	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → (𝑥 / (2↑𝑦)) ∈ ℝ)
151		peano2re 11354	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
152		redivcl 11908	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
153	151, 152	syl3an1 1163	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
154	150, 153	opelxpd 5680	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
155	154	3expb 1120	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0)) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
156	140, 149, 155	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
157	156	rgen2 3178	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ)
158		eqid 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
159	158	fmpo 8050	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ) ↔ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ))
160	157, 159	mpbi 230	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ)
161		frn 6698	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℝ × ℝ))
162	160, 161	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℝ × ℝ)
163	42, 162	sstri 3959	. . . . . . . . . . . . . 14 ⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ × ℝ)
164	53, 163	sstri 3959	. . . . . . . . . . . . 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 11132	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
166		xpss12 5656	. . . . . . . . . . . . . 14 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
167	165, 165, 166	mp2an 692	. . . . . . . . . . . . 13 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
168	164, 167	sstri 3959	. . . . . . . . . . . 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 6707	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℂ × ℂ)) → 𝑓:ℕ⟶(ℂ × ℂ))
170	168, 169	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶(ℂ × ℂ))
171		fco 6715	. . . . . . . . . . 11 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ)
172	139, 170, 171	sylancr 587	. . . . . . . . . 10 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ)
173		nnuz 12843	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
174		1z 12570	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
175	174	a1i 11	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → 1 ∈ ℤ)
176		ffvelcdm 7056	. . . . . . . . . . 11 ⊢ ((((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑛) ∈ ℝ)
177	173, 175, 176	serfre 14003	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ)
178		frn 6698	. . . . . . . . . . 11 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ)
179		ressxr 11225	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ*
180	178, 179	sstrdi 3962	. . . . . . . . . 10 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
181	52, 172, 177, 180	4syl 19	. . . . . . . . 9 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
182		rexr 11227	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℝ → 𝑀 ∈ ℝ*)
183	182	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → 𝑀 ∈ ℝ*)
184		supxrlub 13292	. . . . . . . . 9 ⊢ ((ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* ∧ 𝑀 ∈ ℝ*) → (𝑀 < sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ ∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧))
185	181, 183, 184	syl2anr 597	. . . . . . . 8 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (𝑀 < sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔ ∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧))
186	135, 185	mpbid 232	. . . . . . 7 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧)
187		seqfn 13985	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ≥‘1))
188	174, 187	ax-mp 5	. . . . . . . . 9 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ≥‘1)
189	173	fneq2i 6619	. . . . . . . . 9 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ≥‘1))
190	188, 189	mpbir 231	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ
191		breq2 5114	. . . . . . . . 9 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → (𝑀 < 𝑧 ↔ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
192	191	rexrn 7062	. . . . . . . 8 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ → (∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧 ↔ ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
193	190, 192	ax-mp 5	. . . . . . 7 ⊢ (∃𝑧 ∈ ran seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧 ↔ ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
194	186, 193	sylib 218	. . . . . 6 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
195	60	ffvelcdmda 7059	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
196		0le0 12294	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ≤ 0
197		df-br 5111	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
198	196, 197	mpbi 230	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ ≤
199		0re 11183	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
200		opelxpi 5678	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
201	199, 199, 200	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
202		elin 3933	. . . . . . . . . . . . . . . . 17 ⊢ (⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨0, 0⟩ ∈ ≤ ∧ ⟨0, 0⟩ ∈ (ℝ × ℝ)))
203	198, 201, 202	mpbir2an 711	. . . . . . . . . . . . . . . 16 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
204		ifcl 4537	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
205	195, 203, 204	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
206	205	fmpttd 7090	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
207		df-ov 7393	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0(,)0) = ((,)‘⟨0, 0⟩)
208		iooid 13341	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0(,)0) = ∅
209	207, 208	eqtr3i 2755	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((,)‘⟨0, 0⟩) = ∅
210	209	ineq1i 4182	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = (∅ ∩ ((,)‘(𝑓‘𝑧)))
211		0in 4363	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∩ ((,)‘(𝑓‘𝑧))) = ∅
212	210, 211	eqtri 2753	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅
213	212	olci 866	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅)
214		ineq1 4179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))))
215	214	eqeq1d 2732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
216	215	orbi2d 915	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)))
217		ineq1 4179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((,)‘⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) → (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))))
218	217	eqeq1d 2732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) → ((((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
219	218	orbi2d 915	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) → ((𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)))
220	216, 219	ifboth 4531	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ∧ (𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
221	112, 213, 220	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
222	209	ineq2i 4183	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅)
223		in0 4361	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅) = ∅
224	222, 223	eqtri 2753	. . . . . . . . . . . . . . . . . 18 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅
225	224	olci 866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅)
226		ineq2 4180	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)) → (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))))
227	226	eqeq1d 2732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)) → ((if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
228	227	orbi2d 915	. . . . . . . . . . . . . . . . . 18 ⊢ (((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)) → ((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅)))
229		ineq2 4180	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘⟨0, 0⟩) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)) → (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))))
230	229	eqeq1d 2732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘⟨0, 0⟩) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)) → ((if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
231	230	orbi2d 915	. . . . . . . . . . . . . . . . . 18 ⊢ (((,)‘⟨0, 0⟩) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)) → ((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅)))
232	228, 231	ifboth 4531	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ∧ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
233	221, 225, 232	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
234	233	ralrimivva 3181	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
235		disjeq2 5081	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑚 ∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) → (Disj 𝑚 ∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) ↔ Disj 𝑚 ∈ ℕ if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩))))
236		eleq1w 2812	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑚 → (𝑧 ∈ (1...𝑛) ↔ 𝑚 ∈ (1...𝑛)))
237		fveq2 6861	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑚 → (𝑓‘𝑧) = (𝑓‘𝑚))
238	236, 237	ifbieq1d 4516	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑚 → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩))
239		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))
240		fvex 6874	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓‘𝑚) ∈ V
241		opex 5427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨0, 0⟩ ∈ V
242	240, 241	ifex 4542	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩) ∈ V
243	238, 239, 242	fvmpt 6971	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℕ → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩))
244	243	fveq2d 6865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ → ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) = ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩)))
245		fvif 6877	. . . . . . . . . . . . . . . . . 18 ⊢ ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩))
246	244, 245	eqtrdi 2781	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)))
247	235, 246	mprg 3051	. . . . . . . . . . . . . . . 16 ⊢ (Disj 𝑚 ∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) ↔ Disj 𝑚 ∈ ℕ if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)))
248		eleq1w 2812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (𝑚 ∈ (1...𝑛) ↔ 𝑧 ∈ (1...𝑛)))
249	248, 115	ifbieq1d 4516	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)))
250	249	disjor 5092	. . . . . . . . . . . . . . . 16 ⊢ (Disj 𝑚 ∈ ℕ if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
251	247, 250	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (Disj 𝑚 ∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩))) = ∅))
252	234, 251	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → Disj 𝑚 ∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)))
253		eqid 2730	. . . . . . . . . . . . . 14 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) = seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))
254	206, 252, 253	uniiccvol 25488	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))), ℝ*, < ))
255	254	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) = sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))), ℝ*, < ))
256		rexpssxrxp 11226	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
257	164, 256	sstri 3959	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ* × ℝ*)
258	257, 65	sselid 3947	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℝ* × ℝ*))
259		0xr 11228	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ*
260		opelxpi 5678	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ⟨0, 0⟩ ∈ (ℝ* × ℝ*))
261	259, 259, 260	mp2an 692	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨0, 0⟩ ∈ (ℝ* × ℝ*)
262		ifcl 4537	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑧) ∈ (ℝ* × ℝ*) ∧ ⟨0, 0⟩ ∈ (ℝ* × ℝ*)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ (ℝ* × ℝ*))
263	258, 261, 262	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ (ℝ* × ℝ*))
264		eqidd 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))
265		iccf 13416	. . . . . . . . . . . . . . . . . . . 20 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
266	265	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*)
267	266	feqmptd 6932	. . . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) → ([,]‘𝑚) = ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))
269	263, 264, 267, 268	fmptco 7104	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))) = (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))
270	52, 269	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))) = (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))
271	270	rneqd 5905	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))) = ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))
272	271	unieqd 4887	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))) = ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))
273		peano2nn 12205	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
274	273, 173	eleqtrdi 2839	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ (ℤ≥‘1))
275		fzouzsplit 13662	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘1) → (ℤ≥‘1) = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
276	274, 275	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (ℤ≥‘1) = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
277	173, 276	eqtrid 2777	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ℕ = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
278		nnz 12557	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
279		fzval3 13702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (1...𝑛) = (1..^(𝑛 + 1)))
280	278, 279	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = (1..^(𝑛 + 1)))
281	280	uneq1d 4133	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1))) = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
282	277, 281	eqtr4d 2768	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ℕ = ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1))))
283		fvif 6877	. . . . . . . . . . . . . . . . . 18 ⊢ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩))
284	283	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)))
285	282, 284	iuneq12d 4988	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = ∪ 𝑧 ∈ ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1)))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)))
286		fvex 6874	. . . . . . . . . . . . . . . . 17 ⊢ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) ∈ V
287	286	dfiun3 5936	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))
288		iunxun 5061	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 ∈ ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1)))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)))
289	285, 287, 288	3eqtr3g 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))) = (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩))))
290		iftrue 4497	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘(𝑓‘𝑧)))
291	290	iuneq2i 4980	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))
292	291	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))
293		uznfz 13578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (ℤ≥‘(𝑛 + 1)) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
294	293	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
295		nncn 12201	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
296		ax-1cn 11133	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
297		pncan 11434	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
298	295, 296, 297	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛)
299	298	oveq2d 7406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ → (1...((𝑛 + 1) − 1)) = (1...𝑛))
300	299	eleq2d 2815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → (𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ 𝑧 ∈ (1...𝑛)))
301	300	notbid 318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
302	301	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
303	294, 302	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...𝑛))
304	303	iffalsed 4502	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘⟨0, 0⟩))
305	304	iuneq2dv 4983	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩))
306	292, 305	uneq12d 4135	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩))) = (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)))
307	289, 306	eqtrd 2765	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))) = (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)))
308	272, 307	sylan9eq 2785	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))) = (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)))
309	308	fveq2d 6865	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) = (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩))))
310		xrltso 13108	. . . . . . . . . . . . . . 15 ⊢ < Or ℝ*
311	310	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → < Or ℝ*)
312		elnnuz 12844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
313	312	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ≥‘1))
314	313	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ (ℤ≥‘1))
315		elfznn 13521	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (1...𝑛) → 𝑢 ∈ ℕ)
316	172	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑢 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑢) ∈ ℝ)
317	315, 316	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑢 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑢) ∈ ℝ)
318	317	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑢 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑢) ∈ ℝ)
319		readdcl 11158	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ)
320	319	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ (𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑢 + 𝑣) ∈ ℝ)
321	314, 318, 320	seqcl 13994	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
322	321	rexrd 11231	. . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (1...𝑛) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))
324		iftrue 4497	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ (1...𝑛) → if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩) = (𝑓‘𝑚))
325	238, 324	sylan9eqr 2787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ (1...𝑛) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = (𝑓‘𝑚))
326		elfznn 13521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (1...𝑛) → 𝑚 ∈ ℕ)
327	240	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (1...𝑛) → (𝑓‘𝑚) ∈ V)
328	323, 325, 326, 327	fvmptd 6978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (1...𝑛) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓‘𝑚))
329	328	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓‘𝑚))
330	329	fveq2d 6865	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) = ((abs ∘ − )‘(𝑓‘𝑚)))
331		fvex 6874	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓‘𝑧) ∈ V
332	331, 241	ifex 4542	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ V
333	332, 239	fnmpti 6664	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) Fn ℕ
334		fvco2 6961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)))
335	333, 326, 334	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...𝑛) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)))
336	335	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)))
337		ffn 6691	. . . . . . . . . . . . . . . . . . 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 6961	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚)))
339	337, 326, 338	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚)))
340	330, 336, 339	3eqtr4d 2775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = (((abs ∘ − ) ∘ 𝑓)‘𝑚))
341	340	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = (((abs ∘ − ) ∘ 𝑓)‘𝑚))
342	314, 341	seqfveq 13998	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
343	174	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 1 ∈ ℤ)
344	168, 65	sselid 3947	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℂ × ℂ))
345		0cn 11173	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
346		opelxpi 5678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨0, 0⟩ ∈ (ℂ × ℂ))
347	345, 345, 346	mp2an 692	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨0, 0⟩ ∈ (ℂ × ℂ)
348		ifcl 4537	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓‘𝑧) ∈ (ℂ × ℂ) ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ (ℂ × ℂ))
349	344, 347, 348	sylancl 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ (ℂ × ℂ))
350	349	fmpttd 7090	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)):ℕ⟶(ℂ × ℂ))
351		fco 6715	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)):ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))):ℕ⟶ℝ)
352	139, 350, 351	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))):ℕ⟶ℝ)
353	352	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) ∈ ℝ)
354	173, 343, 353	serfre 14003	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))):ℕ⟶ℝ)
355	354	ffnd 6692	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) Fn ℕ)
356		fnfvelrn 7055	. . . . . . . . . . . . . . . 16 ⊢ ((seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) Fn ℕ ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑛) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))))
357	355, 356	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑛) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))))
358	342, 357	eqeltrrd 2830	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))))
359	354	frnd 6699	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) ⊆ ℝ)
360	359	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) ⊆ ℝ)
361	360	sselda 3949	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))) → 𝑚 ∈ ℝ)
362	321	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
363		readdcl 11158	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑚 + 𝑢) ∈ ℝ)
364	363	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ (𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ)) → (𝑚 + 𝑢) ∈ ℝ)
365		recn 11165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℝ → 𝑚 ∈ ℂ)
366		recn 11165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
367		recn 11165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 ∈ ℝ → 𝑣 ∈ ℂ)
368		addass 11162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
369	365, 366, 367, 368	syl3an 1160	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
370	369	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ (𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
371		nnltp1le 12597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 < 𝑡 ↔ (𝑛 + 1) ≤ 𝑡))
372	371	biimpa 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑛 + 1) ≤ 𝑡)
373	273	nnzd 12563	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℤ)
374		nnz 12557	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ ℤ)
375		eluz 12814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 + 1) ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
376	373, 374, 375	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
377	376	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
378	372, 377	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1)))
379	378	adantlll 718	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1)))
380	313	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑛 ∈ (ℤ≥‘1))
381		simplll 774	. . . . . . . . . . . . . . . . . . . . . . 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 13521	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℕ)
383	381, 382, 353	syl2an 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) ∈ ℝ)
384	364, 370, 379, 380, 383	seqsplit 14007	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) = ((seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡)))
385	342	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑛) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
386		elfzelz 13492	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℤ)
387	386	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
388		0red 11184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
389	273	nnred 12208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
390	389	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
391	386	zred 12645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℝ)
392	391	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
393	273	nngt0d 12242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 ∈ ℕ → 0 < (𝑛 + 1))
394	393	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
395		elfzle1 13495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → (𝑛 + 1) ≤ 𝑚)
396	395	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
397	388, 390, 392, 394, 396	ltletrd 11341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
398		elnnz 12546	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ ℕ ↔ (𝑚 ∈ ℤ ∧ 0 < 𝑚))
399	387, 397, 398	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ)
400	333, 399, 334	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)))
401		eqidd 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))
402		nnre 12200	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
403	402	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 ∈ ℝ)
404	389	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
405	391	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
406	402	ltp1d 12120	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
407	406	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < (𝑛 + 1))
408	395	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
409	403, 404, 405, 407, 408	ltletrd 11341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < 𝑚)
410	409	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → 𝑛 < 𝑚)
411	403, 405	ltnled 11328	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
412		breq1 5113	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑚 = 𝑧 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛))
413	412	equcoms 2020	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 = 𝑚 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛))
414	413	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 = 𝑚 → (¬ 𝑚 ≤ 𝑛 ↔ ¬ 𝑧 ≤ 𝑛))
415	411, 414	sylan9bb 509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → (𝑛 < 𝑚 ↔ ¬ 𝑧 ≤ 𝑛))
416	410, 415	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ≤ 𝑛)
417		elfzle2 13496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ≤ 𝑛)
418	416, 417	nsyl 140	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ∈ (1...𝑛))
419	418	iffalsed 4502	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = ⟨0, 0⟩)
420	386	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
421		0red 11184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
422	393	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
423	421, 404, 405, 422, 408	ltletrd 11341	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
424	420, 423, 398	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ)
425	241	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ⟨0, 0⟩ ∈ V)
426	401, 419, 424, 425	fvmptd 6978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = ⟨0, 0⟩)
427	426	ad4ant14 752	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = ⟨0, 0⟩)
428	427	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) = ((abs ∘ − )‘⟨0, 0⟩))
429	400, 428	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = ((abs ∘ − )‘⟨0, 0⟩))
430		fvco3 6963	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩)))
431	137, 347, 430	mp2an 692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩))
432		df-ov 7393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 − 0) = ( − ‘⟨0, 0⟩)
433		0m0e0 12308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 − 0) = 0
434	432, 433	eqtr3i 2755	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ( − ‘⟨0, 0⟩) = 0
435	434	fveq2i 6864	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (abs‘( − ‘⟨0, 0⟩)) = (abs‘0)
436		abs0 15258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (abs‘0) = 0
437	435, 436	eqtri 2753	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (abs‘( − ‘⟨0, 0⟩)) = 0
438	431, 437	eqtri 2753	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((abs ∘ − )‘⟨0, 0⟩) = 0
439	429, 438	eqtrdi 2781	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = 0)
440		elfzuz 13488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ (ℤ≥‘(𝑛 + 1)))
441		c0ex 11175	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ V
442	441	fvconst2 7181	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ (ℤ≥‘(𝑛 + 1)) → (((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0)
443	440, 442	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → (((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0)
444	443	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0)
445	439, 444	eqtr4d 2768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = (((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚))
446	378, 445	seqfveq 13998	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) = (seq(𝑛 + 1)( + , ((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡))
447		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1))
448	447	ser0 14026	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) → (seq(𝑛 + 1)( + , ((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡) = 0)
449	378, 448	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡) = 0)
450	446, 449	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) = 0)
451	450	adantlll 718	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) = 0)
452	385, 451	oveq12d 7408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡)) = ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) + 0))
453	172	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
454	326, 453	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
455	454	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
456		readdcl 11158	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑚 + 𝑣) ∈ ℝ)
457	456	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑚 + 𝑣) ∈ ℝ)
458	314, 455, 457	seqcl 13994	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
459	458	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
460	459	recnd 11209	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℂ)
461	460	addridd 11381	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) + 0) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
462	452, 461	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡)) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
463	384, 462	eqtrd 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
464	453	ad5ant15 758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
465	326, 464	sylan2 593	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
466	380, 465, 364	seqcl 13994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ)
467	466	leidd 11751	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
468	463, 467	eqbrtrd 5132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
469		elnnuz 12844	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ ℕ ↔ 𝑡 ∈ (ℤ≥‘1))
470	469	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ (ℤ≥‘1))
471	470	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑡 ∈ (ℤ≥‘1))
472		eqidd 2731	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))
473		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 = 𝑚)
474		elfzle1 13495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ (1...𝑡) → 1 ≤ 𝑚)
475	474	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 1 ≤ 𝑚)
476	382	nnred 12208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℝ)
477	476	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℝ)
478		nnre 12200	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ ℝ)
479	478	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ∈ ℝ)
480	402	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑛 ∈ ℝ)
481		elfzle2 13496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ≤ 𝑡)
482	481	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑡)
483		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ≤ 𝑛)
484	477, 479, 480, 482, 483	letrd 11338	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑛)
485		elfzelz 13492	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℤ)
486	278	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ ℤ)
487		elfz 13481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
488	174, 487	mp3an2 1451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
489	485, 486, 488	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
490	475, 484, 489	mpbir2and 713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛))
491	490	ad5ant2345 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛))
492	491	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑚 ∈ (1...𝑛))
493	473, 492	eqeltrd 2829	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 ∈ (1...𝑛))
494		iftrue 4497	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = (𝑓‘𝑧))
495	493, 494	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = (𝑓‘𝑧))
496	237	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → (𝑓‘𝑧) = (𝑓‘𝑚))
497	495, 496	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = (𝑓‘𝑚))
498	382	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℕ)
499	240	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑓‘𝑚) ∈ V)
500	472, 497, 498, 499	fvmptd 6978	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓‘𝑚))
501	500	fveq2d 6865	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) = ((abs ∘ − )‘(𝑓‘𝑚)))
502	333, 382, 334	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...𝑡) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)))
503	502	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = ((abs ∘ − )‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)))
504		simplll 774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
505		fvco3 6963	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚)))
506	504, 382, 505	syl2an 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚)))
507	501, 503, 506	3eqtr4d 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))‘𝑚) = (((abs ∘ − ) ∘ 𝑓)‘𝑚))
508	471, 507	seqfveq 13998	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑡))
509		eluz 12814	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛))
510	374, 278, 509	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 ∈ (ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛))
511	510	biimpar 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ (ℤ≥‘𝑡))
512	511	adantlll 718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ (ℤ≥‘𝑡))
513	504, 326, 453	syl2an 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) ∈ ℝ)
514		elfzelz 13492	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℤ)
515	514	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℤ)
516		0red 11184	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ∈ ℝ)
517		peano2nn 12205	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℕ)
518	517	nnred 12208	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℝ)
519	518	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ∈ ℝ)
520	514	zred 12645	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℝ)
521	520	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℝ)
522	517	nngt0d 12242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ ℕ → 0 < (𝑡 + 1))
523	522	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < (𝑡 + 1))
524		elfzle1 13495	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → (𝑡 + 1) ≤ 𝑚)
525	524	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ≤ 𝑚)
526	516, 519, 521, 523, 525	ltletrd 11341	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < 𝑚)
527	515, 526, 398	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
528	527	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑡 ∈ ℕ ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
529	528	adantlll 718	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
530	170	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ (ℂ × ℂ))
531		ffvelcdm 7056	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) → ( − ‘(𝑓‘𝑚)) ∈ ℂ)
532	137, 530, 531	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → ( − ‘(𝑓‘𝑚)) ∈ ℂ)
533	532	absge0d 15420	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → 0 ≤ (abs‘( − ‘(𝑓‘𝑚))))
534		fvco3 6963	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) → ((abs ∘ − )‘(𝑓‘𝑚)) = (abs‘( − ‘(𝑓‘𝑚))))
535	137, 530, 534	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → ((abs ∘ − )‘(𝑓‘𝑚)) = (abs‘( − ‘(𝑓‘𝑚))))
536	505, 535	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = (abs‘( − ‘(𝑓‘𝑚))))
537	533, 536	breqtrrd 5138	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝑓)‘𝑚))
538	537	ad5ant15 758	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ℕ) → 0 ≤ (((abs ∘ − ) ∘ 𝑓)‘𝑚))
539	529, 538	syldan 591	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ≤ (((abs ∘ − ) ∘ 𝑓)‘𝑚))
540	471, 512, 513, 539	sermono 14006	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
541	508, 540	eqbrtrd 5132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
542	402	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑛 ∈ ℝ)
543	478	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℝ)
544	468, 541, 542, 543	ltlecasei 11289	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
545	544	ralrimiva 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), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
546		breq1 5113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) → (𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
547	546	ralrn 7063	. . . . . . . . . . . . . . . . . . 19 ⊢ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) Fn ℕ → (∀𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
548	355, 547	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (∀𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
549	548	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (∀𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
550	545, 549	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∀𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
551	550	r19.21bi 3230	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))) → 𝑚 ≤ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
552	361, 362, 551	lensymd 11332	. . . . . . . . . . . . . 14 ⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))) → ¬ (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) < 𝑚)
553	311, 322, 358, 552	supmax 9426	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))), ℝ*, < ) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
554	52, 553	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → sup(ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))), ℝ*, < ) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛))
555	255, 309, 554	3eqtr3rd 2774	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) = (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩))))
556		elfznn 13521	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ∈ ℕ)
557	164, 65	sselid 3947	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℝ × ℝ))
558		1st2nd2 8010	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (𝑓‘𝑧) = ⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩)
559	558	fveq2d 6865	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) = ([,]‘⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩))
560		df-ov 7393	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) = ([,]‘⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩)
561	559, 560	eqtr4di 2783	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) = ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))))
562		xp1st 8003	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (1st ‘(𝑓‘𝑧)) ∈ ℝ)
563		xp2nd 8004	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (2nd ‘(𝑓‘𝑧)) ∈ ℝ)
564		iccssre 13397	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑧)) ∈ ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ)
565	562, 563, 564	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ)
566	561, 565	eqsstrd 3984	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) ⊆ ℝ)
567	557, 566	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ⊆ ℝ)
568	52, 556, 567	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ⊆ ℝ)
569	568	ralrimiva 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)([,]‘(𝑓‘𝑧)) ⊆ ℝ)
570		iunss 5012	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ)
571	569, 570	sylibr 234	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ)
572	571	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ)
573		uzid 12815	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℤ → (𝑛 + 1) ∈ (ℤ≥‘(𝑛 + 1)))
574		ne0i 4307	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘(𝑛 + 1)) → (ℤ≥‘(𝑛 + 1)) ≠ ∅)
575		iunconst 4968	. . . . . . . . . . . . . . . 16 ⊢ ((ℤ≥‘(𝑛 + 1)) ≠ ∅ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) = ([,]‘⟨0, 0⟩))
576	373, 573, 574, 575	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) = ([,]‘⟨0, 0⟩))
577		iccid 13358	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ* → (0[,]0) = {0})
578	259, 577	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (0[,]0) = {0}
579		df-ov 7393	. . . . . . . . . . . . . . . 16 ⊢ (0[,]0) = ([,]‘⟨0, 0⟩)
580	578, 579	eqtr3i 2755	. . . . . . . . . . . . . . 15 ⊢ {0} = ([,]‘⟨0, 0⟩)
581	576, 580	eqtr4di 2783	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) = {0})
582		snssi 4775	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
583	199, 582	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ ℝ
584	581, 583	eqsstrdi 3994	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) ⊆ ℝ)
585	584	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) ⊆ ℝ)
586	581	fveq2d 6865	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = (vol*‘{0}))
587	586	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = (vol*‘{0}))
588		ovolsn 25403	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ → (vol*‘{0}) = 0)
589	199, 588	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol*‘{0}) = 0
590	587, 589	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = 0)
591		ovolunnul 25408	. . . . . . . . . . . 12 ⊢ ((∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ ∧ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) ⊆ ℝ ∧ (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = 0) → (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩))) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))
592	572, 585, 590, 591	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩))) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))
593	555, 592	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))
594	593	breq2d 5122	. . . . . . . . 9 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ↔ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))
595	594	biimpd 229	. . . . . . . 8 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))
596	595	reximdva 3147	. . . . . . 7 ⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))
597	596	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))
598	194, 597	mpd 15	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))
599		fzfi 13944	. . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin
600		icccld 24661	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑧)) ∈ ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
601	562, 563, 600	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
602	561, 601	eqeltrd 2829	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
603	557, 602	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
604	556, 603	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
605	604	ralrimiva 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)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
606		uniretop 24657	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
607	606	iuncld 22939	. . . . . . . . . 10 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (1...𝑛) ∈ Fin ∧ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,)))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
608	1, 599, 605, 607	mp3an12i 1467	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
609	608	adantr 480	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
610		fveq2 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑧) → ([,]‘𝑏) = ([,]‘(𝑓‘𝑧)))
611	610	sseq1d 3981	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑓‘𝑧) → (([,]‘𝑏) ⊆ 𝐴 ↔ ([,]‘(𝑓‘𝑧)) ⊆ 𝐴))
612	611	elrab 3662	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ↔ ((𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∧ ([,]‘(𝑓‘𝑧)) ⊆ 𝐴))
613	612	simprbi 496	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
614	65, 73, 613	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
615	556, 614	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
616	615	ralrimiva 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)([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
617		iunss 5012	. . . . . . . . . 10 ⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
618	616, 617	sylibr 234	. . . . . . . . 9 ⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
619	618	adantr 480	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
620		simprr 772	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))
621		sseq1 3975	. . . . . . . . . 10 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑠 ⊆ 𝐴 ↔ ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴))
622		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (vol*‘𝑠) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))
623	622	breq2d 5122	. . . . . . . . . 10 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))
624	621, 623	anbi12d 632	. . . . . . . . 9 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → ((𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)) ↔ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))))
625	624	rspcev 3591	. . . . . . . 8 ⊢ ((∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))) ∧ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
626	609, 619, 620, 625	syl12anc 836	. . . . . . 7 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
627	52, 626	sylan 580	. . . . . 6 ⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
628	627	adantll 714	. . . . 5 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
629	598, 628	rexlimddv 3141	. . . 4 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
630	629	adantlr 715	. . 3 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
631	17, 630	exlimddv 1935	. 2 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
632	15, 631	pm2.61dane 3013	1 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  {crab 3408  Vcvv 3450   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  ifcif 4491  𝒫 cpw 4566  {csn 4592  ⟨cop 4598  ∪ cuni 4874  ∪ ciun 4958  Disj wdisj 5077   class class class wbr 5110   ↦ cmpt 5191   Or wor 5548   × cxp 5639  ran crn 5642   “ cima 5644   ∘ ccom 5645   Fn wfn 6509  ⟶wf 6510  –1-1→wf1 6511  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1^st c1st 7969  2^nd c2nd 7970  Fincfn 8921  supcsup 9398  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216   − cmin 11412   / cdiv 11842  ℕcn 12193  2c2 12248  ℕ₀cn0 12449  ℤcz 12536  ℤ_≥cuz 12800  (,)cioo 13313  [,]cicc 13316  ...cfz 13475  ..^cfzo 13622  seqcseq 13973  ↑cexp 14033  abscabs 15207  topGenctg 17407  Topctop 22787  Clsdccld 22910  vol*covol 25370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-disj 5078  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-acn 9902  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ioo 13317  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-rest 17392  df-topgen 17413  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-top 22788  df-topon 22805  df-bases 22840  df-cld 22913  df-cmp 23281  df-conn 23306  df-ovol 25372  df-vol 25373

This theorem is referenced by:  mblfinlem4  37661  ismblfin  37662