Theorem mblfinlem2 38025

Description: Lemma for ismblfin 38028, 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 24744	. . . 4 ⊢ (topGen‘ran (,)) ∈ Top
2		0cld 23021	. . . 4 ⊢ ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
3	1, 2	ax-mp 5	. . 3 ⊢ ∅ ∈ (Clsd‘(topGen‘ran (,)))
4		simpl3 1200	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘𝐴))
5		fveq2 6827	. . . . . 6 ⊢ (𝐴 = ∅ → (vol*‘𝐴) = (vol*‘∅))
6	5	adantl 482	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (vol*‘𝐴) = (vol*‘∅))
7	4, 6	breqtrd 5098	. . . 4 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘∅))
8		0ss 4328	. . . 4 ⊢ ∅ ⊆ 𝐴
9	7, 8	jctil 524	. . 3 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅)))
10		sseq1 3940	. . . . 5 ⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
11		fveq2 6827	. . . . . 6 ⊢ (𝑠 = ∅ → (vol*‘𝑠) = (vol*‘∅))
12	11	breq2d 5084	. . . . 5 ⊢ (𝑠 = ∅ → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∅)))
13	10, 12	anbi12d 638	. . . 4 ⊢ (𝑠 = ∅ → ((𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)) ↔ (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅))))
14	13	rspcev 3560	. . 3 ⊢ ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
15	3, 9, 14	sylancr 593	. 2 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
16		mblfinlem1 38024	. . . 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 1192	. . 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 1200	. . . . . . . . 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 6774	. . . . . . . . . . . . . . 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 6205	. . . . . . . . . . . . . . . . 17 ⊢ ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
21		forn 6742	. . . . . . . . . . . . . . . . . 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 6012	. . . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . 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 4851	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 7363	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦)))
28		oveq1 7363	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1))
29	28	oveq1d 7371	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦)))
30	27, 29	opeq12d 4812	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩)
31		oveq2 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣))
32	31	oveq2d 7372	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣)))
33	31	oveq2d 7372	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣)))
34	32, 33	opeq12d 4812	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))⟩)
35	30, 34	cbvmpov 7451	. . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧))
37	36	sseq1d 3946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐)))
38		eqeq1 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (𝑎 = 𝑐 ↔ 𝑧 = 𝑐))
39	37, 38	imbi12d 345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
40	39	ralbidv 3162	. . . . . . . . . . . . . . . 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 3401	. . . . . . . . . . . . . . 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 4011	. . . . . . . . . . . . . . . 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 25584	. . . . . . . . . . . . . 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 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)}) → ∪ ([,] “ {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 2777	. . . . . . . . . . . 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 38023	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
48	47	3ad2ant1 1139	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 2774	. . . . . . . . . . 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 6831	. . . . . . . . . 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 6767	. . . . . . . . . . . . 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 4011	. . . . . . . . . . . . . 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 25576	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
55		frn 6662	. . . . . . . . . . . . . . . 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 3924	. . . . . . . . . . . . . 14 ⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ( ≤ ∩ (ℝ × ℝ))
58	53, 57	sstri 3924	. . . . . . . . . . . . 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 6671	. . . . . . . . . . . . 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 592	. . . . . . . . . . . 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 3924	. . . . . . . . . . . . . . . . . . . 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 7022	. . . . . . . . . . . . . . . . . . . 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 3913	. . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . 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 7022	. . . . . . . . . . . . . . . . . . . 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 3913	. . . . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . . 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 25581	. . . . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . 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 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)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
71		df-3or 1093	. . . . . . . . . . . . . . . 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 219	. . . . . . . . . . . . . . 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 3625	. . . . . . . . . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑚) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑚)))
75	74	sseq1d 3946	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑚) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐)))
76		eqeq1 2743	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑚) → (𝑎 = 𝑐 ↔ (𝑓‘𝑚) = 𝑐))
77	75, 76	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝑓‘𝑚) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)))
78	77	ralbidv 3162	. . . . . . . . . . . . . . . . . . . . . . . 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 3629	. . . . . . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑧) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑧)))
82	81	sseq2d 3947	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑧) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧))))
83		eqeq2 2751	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑧) → ((𝑓‘𝑚) = 𝑐 ↔ (𝑓‘𝑚) = (𝑓‘𝑧)))
84	82, 83	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (𝑓‘𝑧) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧))))
85	84	rspcva 3558	. . . . . . . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . . . . . . . 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 3625	. . . . . . . . . . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = (𝑓‘𝑧) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑧)))
89	88	sseq1d 3946	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑧) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐)))
90		eqeq1 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑧) → (𝑎 = 𝑐 ↔ (𝑓‘𝑧) = 𝑐))
91	89, 90	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑧) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)))
92	91	ralbidv 3162	. . . . . . . . . . . . . . . . . . . . . . . . 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 3629	. . . . . . . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 = (𝑓‘𝑚) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑚)))
96	95	sseq2d 3947	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑚) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))))
97		eqeq2 2751	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑚) → ((𝑓‘𝑧) = 𝑐 ↔ (𝑓‘𝑧) = (𝑓‘𝑚)))
98	96, 97	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑚) → ((([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚))))
99	98	rspcva 3558	. . . . . . . . . . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑧) = (𝑓‘𝑚) ↔ (𝑓‘𝑚) = (𝑓‘𝑧))
102	100, 101	imbitrdi 252	. . . . . . . . . . . . . . . . . . . . 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 865	. . . . . . . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . . . . . . 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 684	. . . . . . . . . . . . . . . . . 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 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)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧)))
107		f1of1 6766	. . . . . . . . . . . . . . . . . 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 7200	. . . . . . . . . . . . . . . . . 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 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)) → ((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 973	. . . . . . . . . . . . . . 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 3182	. . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → (𝑚 = 𝑝 ↔ 𝑧 = 𝑝))
115		2fveq3 6832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → ((,)‘(𝑓‘𝑚)) = ((,)‘(𝑓‘𝑧)))
116	115	ineq1d 4148	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))))
117	116	eqeq1d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅ ↔ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
118	114, 117	orbi12d 924	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑧 → ((𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
119	118	ralbidv 3162	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑧 → (∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
120	119	cbvralvw 3217	. . . . . . . . . . . . . 14 ⊢ (∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
121		eqeq2 2751	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑝 → (𝑚 = 𝑧 ↔ 𝑚 = 𝑝))
122		2fveq3 6832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑝 → ((,)‘(𝑓‘𝑧)) = ((,)‘(𝑓‘𝑝)))
123	122	ineq2d 4149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑝 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))))
124	123	eqeq1d 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑝 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
125	121, 124	orbi12d 924	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑝 → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
126	125	cbvralvw 3217	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
127	126	ralbii 3085	. . . . . . . . . . . . . 14 ⊢ (∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
128	122	disjor 5054	. . . . . . . . . . . . . 14 ⊢ (Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
129	120, 127, 128	3bitr4ri 305	. . . . . . . . . . . . 13 ⊢ (Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
130	113, 129	sylibr 235	. . . . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
132	60, 130, 131	uniiccvol 25565	. . . . . . . . . . 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 482	. . . . . . . . . 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 2776	. . . . . . . . 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 5098	. . . . . . . 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 15291	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
137		subf 11386	. . . . . . . . . . . 12 ⊢ − :(ℂ × ℂ)⟶ℂ
138		fco 6679	. . . . . . . . . . . 12 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
139	136, 137, 138	mp2an 698	. . . . . . . . . . 11 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
140		zre 12519	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
141		2re 12246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
142		reexpcl 14031	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℝ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℝ)
143	141, 142	mpan 696	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℝ)
144		2cn 12247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℂ
145		2ne0 12276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
146		nn0z 12539	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ)
147		expne0i 14047	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0)
148	144, 145, 146, 147	mp3an12i 1473	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → (2↑𝑦) ≠ 0)
149	143, 148	jca 516	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0))
150		redivcl 11865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → (𝑥 / (2↑𝑦)) ∈ ℝ)
151		peano2re 11310	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
152		redivcl 11865	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
153	151, 152	syl3an1 1169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
154	150, 153	opelxpd 5657	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
155	154	3expb 1126	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0)) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
156	140, 149, 155	syl2an 602	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
157	156	rgen2 3179	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ)
158		eqid 2739	. . . . . . . . . . . . . . . . . 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 8010	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ)
161		frn 6662	. . . . . . . . . . . . . . . 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 3924	. . . . . . . . . . . . . 14 ⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ × ℝ)
164	53, 163	sstri 3924	. . . . . . . . . . . . 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 11086	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
166		xpss12 5633	. . . . . . . . . . . . . 14 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
167	165, 165, 166	mp2an 698	. . . . . . . . . . . . 13 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
168	164, 167	sstri 3924	. . . . . . . . . . . 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 6671	. . . . . . . . . . . 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 697	. . . . . . . . . . 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 6679	. . . . . . . . . . 11 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ)
172	139, 170, 171	sylancr 593	. . . . . . . . . 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 12818	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
174		1z 12548	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
175	174	a1i 11	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → 1 ∈ ℤ)
176		ffvelcdm 7022	. . . . . . . . . . 11 ⊢ ((((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑛) ∈ ℝ)
177	173, 175, 176	serfre 13984	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ)
178		frn 6662	. . . . . . . . . . 11 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ)
179		ressxr 11180	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ*
180	178, 179	sstrdi 3927	. . . . . . . . . 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 11182	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℝ → 𝑀 ∈ ℝ*)
183	182	3ad2ant2 1140	. . . . . . . . 9 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → 𝑀 ∈ ℝ*)
184		supxrlub 13268	. . . . . . . . 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 603	. . . . . . . 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 233	. . . . . . 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 13966	. . . . . . . . . 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 6583	. . . . . . . . 9 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ≥‘1))
190	188, 189	mpbir 232	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ
191		breq2 5076	. . . . . . . . 9 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → (𝑀 < 𝑧 ↔ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
192	191	rexrn 7028	. . . . . . . 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 219	. . . . . 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 7025	. . . . . . . . . . . . . . . 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 12273	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ≤ 0
197		df-br 5073	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
198	196, 197	mpbi 231	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ ≤
199		0re 11137	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
200		opelxpi 5655	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
201	199, 199, 200	mp2an 698	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
202		elin 3899	. . . . . . . . . . . . . . . . 17 ⊢ (⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨0, 0⟩ ∈ ≤ ∧ ⟨0, 0⟩ ∈ (ℝ × ℝ)))
203	198, 201, 202	mpbir2an 717	. . . . . . . . . . . . . . . 16 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
204		ifcl 4500	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
205	195, 203, 204	sylancl 592	. . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . 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 7359	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0(,)0) = ((,)‘⟨0, 0⟩)
208		iooid 13317	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0(,)0) = ∅
209	207, 208	eqtr3i 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((,)‘⟨0, 0⟩) = ∅
210	209	ineq1i 4145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = (∅ ∩ ((,)‘(𝑓‘𝑧)))
211		0in 4325	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∩ ((,)‘(𝑓‘𝑧))) = ∅
212	210, 211	eqtri 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅
213	212	olci 872	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅)
214		ineq1 4142	. . . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . 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 921	. . . . . . . . . . . . . . . . . . 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 4142	. . . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . 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 921	. . . . . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅)
223		in0 4323	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅) = ∅
224	222, 223	eqtri 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅
225	224	olci 872	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅)
226		ineq2 4143	. . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . 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 921	. . . . . . . . . . . . . . . . . 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 4143	. . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . 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 921	. . . . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . 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 3182	. . . . . . . . . . . . . . 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 5043	. . . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑚 → (𝑧 ∈ (1...𝑛) ↔ 𝑚 ∈ (1...𝑛)))
237		fveq2 6827	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑚 → (𝑓‘𝑧) = (𝑓‘𝑚))
238	236, 237	ifbieq1d 4479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑚 → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩))
239		eqid 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))
240		fvex 6840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓‘𝑚) ∈ V
241		opex 5403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨0, 0⟩ ∈ V
242	240, 241	ifex 4505	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩) ∈ V
243	238, 239, 242	fvmpt 6935	. . . . . . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ → ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) = ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩)))
245		fvif 6843	. . . . . . . . . . . . . . . . . 18 ⊢ ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩))
246	244, 245	eqtrdi 2790	. . . . . . . . . . . . . . . . 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 3059	. . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (𝑚 ∈ (1...𝑛) ↔ 𝑧 ∈ (1...𝑛)))
249	248, 115	ifbieq1d 4479	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)))
250	249	disjor 5054	. . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 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 25565	. . . . . . . . . . . . 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 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) → (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 11181	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
257	164, 256	sstri 3924	. . . . . . . . . . . . . . . . . . . 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 3913	. . . . . . . . . . . . . . . . . . 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 11183	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ*
260		opelxpi 5655	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ⟨0, 0⟩ ∈ (ℝ* × ℝ*))
261	259, 259, 260	mp2an 698	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨0, 0⟩ ∈ (ℝ* × ℝ*)
262		ifcl 4500	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑧) ∈ (ℝ* × ℝ*) ∧ ⟨0, 0⟩ ∈ (ℝ* × ℝ*)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ (ℝ* × ℝ*))
263	258, 261, 262	sylancl 592	. . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . 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 13392	. . . . . . . . . . . . . . . . . . . 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 6895	. . . . . . . . . . . . . . . . . 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 6827	. . . . . . . . . . . . . . . . . 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 7071	. . . . . . . . . . . . . . . . 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 5880	. . . . . . . . . . . . . . 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 4851	. . . . . . . . . . . . . 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 12177	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
274	273, 173	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ (ℤ≥‘1))
275		fzouzsplit 13640	. . . . . . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ℕ = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
278		nnz 12536	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
279		fzval3 13680	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (1...𝑛) = (1..^(𝑛 + 1)))
280	278, 279	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = (1..^(𝑛 + 1)))
281	280	uneq1d 4097	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1))) = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
282	277, 281	eqtr4d 2777	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ℕ = ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1))))
283		fvif 6843	. . . . . . . . . . . . . . . . . 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 4951	. . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . 17 ⊢ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) ∈ V
287	286	dfiun3 5912	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))
288		iunxun 5023	. . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . 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 4460	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘(𝑓‘𝑧)))
291	290	iuneq2i 4943	. . . . . . . . . . . . . . . . 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 13555	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (ℤ≥‘(𝑛 + 1)) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
294	293	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
295		nncn 12173	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
296		ax-1cn 11087	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
297		pncan 11390	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
298	295, 296, 297	sylancl 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛)
299	298	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ → (1...((𝑛 + 1) − 1)) = (1...𝑛))
300	299	eleq2d 2825	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → (𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ 𝑧 ∈ (1...𝑛)))
301	300	notbid 319	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
302	301	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
303	294, 302	mpbid 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...𝑛))
304	303	iffalsed 4465	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘⟨0, 0⟩))
305	304	iuneq2dv 4946	. . . . . . . . . . . . . . . 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 4099	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . 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 6831	. . . . . . . . . . . 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 13083	. . . . . . . . . . . . . . 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 12819	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
313	312	biimpi 217	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ≥‘1))
314	313	adantl 482	. . . . . . . . . . . . . . . 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 13498	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (1...𝑛) → 𝑢 ∈ ℕ)
316	172	ffvelcdmda 7025	. . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . 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 721	. . . . . . . . . . . . . . . 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 11112	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ)
320	319	adantl 482	. . . . . . . . . . . . . . . 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 13975	. . . . . . . . . . . . . . 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 11186	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . 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 4460	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ (1...𝑛) → if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩) = (𝑓‘𝑚))
325	238, 324	sylan9eqr 2796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ (1...𝑛) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = (𝑓‘𝑚))
326		elfznn 13498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (1...𝑛) → 𝑚 ∈ ℕ)
327	240	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (1...𝑛) → (𝑓‘𝑚) ∈ V)
328	323, 325, 326, 327	fvmptd 6943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (1...𝑛) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓‘𝑚))
329	328	adantl 482	. . . . . . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓‘𝑧) ∈ V
332	331, 241	ifex 4505	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ V
333	332, 239	fnmpti 6628	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) Fn ℕ
334		fvco2 6924	. . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 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 6655	. . . . . . . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚)))
339	337, 326, 338	syl2an 602	. . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . 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 721	. . . . . . . . . . . . . . . 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 13979	. . . . . . . . . . . . . . 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 3913	. . . . . . . . . . . . . . . . . . . . . 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 11127	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
346		opelxpi 5655	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨0, 0⟩ ∈ (ℂ × ℂ))
347	345, 345, 346	mp2an 698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨0, 0⟩ ∈ (ℂ × ℂ)
348		ifcl 4500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓‘𝑧) ∈ (ℂ × ℂ) ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ (ℂ × ℂ))
349	344, 347, 348	sylancl 592	. . . . . . . . . . . . . . . . . . . . 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 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 ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)):ℕ⟶(ℂ × ℂ))
351		fco 6679	. . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . 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 7025	. . . . . . . . . . . . . . . . . 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 13984	. . . . . . . . . . . . . . . . 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 6656	. . . . . . . . . . . . . . . 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 7021	. . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . 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 6663	. . . . . . . . . . . . . . . . 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 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) → ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))) ⊆ ℝ)
361	360	sselda 3915	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 11112	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑚 + 𝑢) ∈ ℝ)
364	363	adantl 482	. . . . . . . . . . . . . . . . . . . . . 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 11119	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℝ → 𝑚 ∈ ℂ)
366		recn 11119	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
367		recn 11119	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 ∈ ℝ → 𝑣 ∈ ℂ)
368		addass 11116	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
369	365, 366, 367, 368	syl3an 1166	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
370	369	adantl 482	. . . . . . . . . . . . . . . . . . . . . 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 12576	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 < 𝑡 ↔ (𝑛 + 1) ≤ 𝑡))
372	371	biimpa 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑛 + 1) ≤ 𝑡)
373	273	nnzd 12541	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℤ)
374		nnz 12536	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ ℤ)
375		eluz 12793	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 + 1) ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
376	373, 374, 375	syl2an 602	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
377	376	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
378	372, 377	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1)))
379	378	adantlll 724	. . . . . . . . . . . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . 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 13498	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℕ)
383	381, 382, 353	syl2an 602	. . . . . . . . . . . . . . . . . . . . . 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 13988	. . . . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . . . 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 13469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℤ)
387	386	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
388		0red 11138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
389	273	nnred 12180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
390	389	ad3antrrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
391	386	zred 12624	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℝ)
392	391	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
393	273	nngt0d 12217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 ∈ ℕ → 0 < (𝑛 + 1))
394	393	ad3antrrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
395		elfzle1 13472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → (𝑛 + 1) ≤ 𝑚)
396	395	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
397	388, 390, 392, 394, 396	ltletrd 11297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
398		elnnz 12525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ ℕ ↔ (𝑚 ∈ ℤ ∧ 0 < 𝑚))
399	387, 397, 398	sylanbrc 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ)
400	333, 399, 334	sylancr 593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
403	402	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 ∈ ℝ)
404	389	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
405	391	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
406	402	ltp1d 12077	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
407	406	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < (𝑛 + 1))
408	395	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
409	403, 404, 405, 407, 408	ltletrd 11297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < 𝑚)
410	409	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → 𝑛 < 𝑚)
411	403, 405	ltnled 11284	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
412		breq1 5075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑚 = 𝑧 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛))
413	412	equcoms 2027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 = 𝑚 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛))
414	413	notbid 319	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 = 𝑚 → (¬ 𝑚 ≤ 𝑛 ↔ ¬ 𝑧 ≤ 𝑛))
415	411, 414	sylan9bb 514	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → (𝑛 < 𝑚 ↔ ¬ 𝑧 ≤ 𝑛))
416	410, 415	mpbid 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ≤ 𝑛)
417		elfzle2 13473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = ⟨0, 0⟩)
420	386	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
421		0red 11138	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
422	393	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
423	421, 404, 405, 422, 408	ltletrd 11297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
424	420, 423, 398	sylanbrc 589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6943	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = ⟨0, 0⟩)
427	426	ad4ant14 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6927	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩)))
431	137, 347, 430	mp2an 698	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩))
432		df-ov 7359	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 − 0) = ( − ‘⟨0, 0⟩)
433		0m0e0 12287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 − 0) = 0
434	432, 433	eqtr3i 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ( − ‘⟨0, 0⟩) = 0
435	434	fveq2i 6830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (abs‘( − ‘⟨0, 0⟩)) = (abs‘0)
436		abs0 15238	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (abs‘0) = 0
437	435, 436	eqtri 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (abs‘( − ‘⟨0, 0⟩)) = 0
438	431, 437	eqtri 2762	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((abs ∘ − )‘⟨0, 0⟩) = 0
439	429, 438	eqtrdi 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ (ℤ≥‘(𝑛 + 1)))
441		c0ex 11129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ V
442	441	fvconst2 7148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0)
445	439, 444	eqtr4d 2777	. . . . . . . . . . . . . . . . . . . . . . . . . 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 13979	. . . . . . . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1))
448	447	ser0 14007	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))))‘𝑡) = 0)
452	385, 451	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . 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 7025	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 721	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11112	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑚 + 𝑣) ∈ ℝ)
457	456	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 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 13975	. . . . . . . . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . . . . 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 11164	. . . . . . . . . . . . . . . . . . . . . . 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 11337	. . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . 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 764	. . . . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . . . 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 13975	. . . . . . . . . . . . . . . . . . . . 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 11707	. . . . . . . . . . . . . . . . . . . 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 5094	. . . . . . . . . . . . . . . . . . 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 12819	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ ℕ ↔ 𝑡 ∈ (ℤ≥‘1))
470	469	biimpi 217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ (ℤ≥‘1))
471	470	ad2antlr 733	. . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ (1...𝑡) → 1 ≤ 𝑚)
475	474	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 1 ≤ 𝑚)
476	382	nnred 12180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℝ)
477	476	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℝ)
478		nnre 12172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ ℝ)
479	478	ad3antlr 737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ∈ ℝ)
480	402	ad3antrrr 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑛 ∈ ℝ)
481		elfzle2 13473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ≤ 𝑡)
482	481	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑡)
483		simplr 774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ≤ 𝑛)
484	477, 479, 480, 482, 483	letrd 11294	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑛)
485		elfzelz 13469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℤ)
486	278	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ ℤ)
487		elfz 13458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
488	174, 487	mp3an2 1457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
489	485, 486, 488	syl2anr 603	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
490	475, 484, 489	mpbir2and 719	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛))
491	490	ad5ant2345 1378	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4460	. . . . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . . 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 6943	. . . . . . . . . . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . 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 6927	. . . . . . . . . . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . . . . . 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 13979	. . . . . . . . . . . . . . . . . . . 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 12793	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛))
510	374, 278, 509	syl2anr 603	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 ∈ (ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛))
511	510	biimpar 478	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ (ℤ≥‘𝑡))
512	511	adantlll 724	. . . . . . . . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . . . . . . . . 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 13469	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℤ)
515	514	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℤ)
516		0red 11138	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ∈ ℝ)
517		peano2nn 12177	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℕ)
518	517	nnred 12180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℝ)
519	518	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ∈ ℝ)
520	514	zred 12624	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℝ)
521	520	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℝ)
522	517	nngt0d 12217	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ ℕ → 0 < (𝑡 + 1))
523	522	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < (𝑡 + 1))
524		elfzle1 13472	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → (𝑡 + 1) ≤ 𝑚)
525	524	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ≤ 𝑚)
526	516, 519, 521, 523, 525	ltletrd 11297	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < 𝑚)
527	515, 526, 398	sylanbrc 589	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
528	527	adantlr 721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑡 ∈ ℕ ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
529	528	adantlll 724	. . . . . . . . . . . . . . . . . . . . . 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 7025	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7022	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) → ( − ‘(𝑓‘𝑚)) ∈ ℂ)
532	137, 530, 531	sylancr 593	. . . . . . . . . . . . . . . . . . . . . . . . 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 15400	. . . . . . . . . . . . . . . . . . . . . . . 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 6927	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) → ((abs ∘ − )‘(𝑓‘𝑚)) = (abs‘( − ‘(𝑓‘𝑚))))
535	137, 530, 534	sylancr 593	. . . . . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . . . . 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 5100	. . . . . . . . . . . . . . . . . . . . . . 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 764	. . . . . . . . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . . . . . . . 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 13987	. . . . . . . . . . . . . . . . . . . 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 5094	. . . . . . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . 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 11245	. . . . . . . . . . . . . . . . . 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 3131	. . . . . . . . . . . . . . . . 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 5075	. . . . . . . . . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . 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 3231	. . . . . . . . . . . . . . 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 11288	. . . . . . . . . . . . . 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 9371	. . . . . . . . . . . . 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 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)} ∧ 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 2783	. . . . . . . . . . 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 13498	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ∈ ℕ)
557	164, 65	sselid 3913	. . . . . . . . . . . . . . . . 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 7970	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (𝑓‘𝑧) = ⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩)
559	558	fveq2d 6831	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) = ([,]‘⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩))
560		df-ov 7359	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) = ([,]‘⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩)
561	559, 560	eqtr4di 2792	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) = ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))))
562		xp1st 7963	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (1st ‘(𝑓‘𝑧)) ∈ ℝ)
563		xp2nd 7964	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (2nd ‘(𝑓‘𝑧)) ∈ ℝ)
564		iccssre 13373	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑧)) ∈ ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ)
565	562, 563, 564	syl2anc 590	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ)
566	561, 565	eqsstrd 3949	. . . . . . . . . . . . . . . . 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 602	. . . . . . . . . . . . . . 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 3131	. . . . . . . . . . . . . 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 4974	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ)
571	569, 570	sylibr 235	. . . . . . . . . . . . 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 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 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ)
573		uzid 12794	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℤ → (𝑛 + 1) ∈ (ℤ≥‘(𝑛 + 1)))
574		ne0i 4269	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘(𝑛 + 1)) → (ℤ≥‘(𝑛 + 1)) ≠ ∅)
575		iunconst 4931	. . . . . . . . . . . . . . . 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 13334	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ* → (0[,]0) = {0})
578	259, 577	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (0[,]0) = {0}
579		df-ov 7359	. . . . . . . . . . . . . . . 16 ⊢ (0[,]0) = ([,]‘⟨0, 0⟩)
580	578, 579	eqtr3i 2764	. . . . . . . . . . . . . . 15 ⊢ {0} = ([,]‘⟨0, 0⟩)
581	576, 580	eqtr4di 2792	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) = {0})
582		snssi 4717	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
583	199, 582	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ ℝ
584	581, 583	eqsstrdi 3959	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) ⊆ ℝ)
585	584	adantl 482	. . . . . . . . . . . 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 6831	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = (vol*‘{0}))
587	586	adantl 482	. . . . . . . . . . . . 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 25480	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ → (vol*‘{0}) = 0)
589	199, 588	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol*‘{0}) = 0
590	587, 589	eqtrdi 2790	. . . . . . . . . . . 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 25485	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 2774	. . . . . . . . . 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 5084	. . . . . . . . 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 230	. . . . . . . 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 3152	. . . . . . 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 482	. . . . . 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 13925	. . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin
600		icccld 24749	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑧)) ∈ ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
601	562, 563, 600	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
602	561, 601	eqeltrd 2839	. . . . . . . . . . . . 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 599	. . . . . . . . . . 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 3131	. . . . . . . . . 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 24745	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
607	606	iuncld 23028	. . . . . . . . . 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 1473	. . . . . . . . 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 481	. . . . . . . 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 6827	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑧) → ([,]‘𝑏) = ([,]‘(𝑓‘𝑧)))
611	610	sseq1d 3946	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑓‘𝑧) → (([,]‘𝑏) ⊆ 𝐴 ↔ ([,]‘(𝑓‘𝑧)) ⊆ 𝐴))
612	611	elrab 3629	. . . . . . . . . . . . . 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 498	. . . . . . . . . . . . 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 599	. . . . . . . . . . 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 3131	. . . . . . . . . 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 4974	. . . . . . . . . 10 ⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
618	616, 617	sylibr 235	. . . . . . . . 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 481	. . . . . . . 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 778	. . . . . . . 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 3940	. . . . . . . . . 10 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑠 ⊆ 𝐴 ↔ ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴))
622		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (vol*‘𝑠) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))
623	622	breq2d 5084	. . . . . . . . . 10 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))
624	621, 623	anbi12d 638	. . . . . . . . 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 3560	. . . . . . . 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 842	. . . . . . 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 586	. . . . . 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 720	. . . . 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 3146	. . . 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 721	. . 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 1942	. 2 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
632	15, 631	pm2.61dane 3021	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 207   ∧ wa 396   ∨ wo 853   ∨ w3o 1091   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  {crab 3391  Vcvv 3431   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  ifcif 4454  𝒫 cpw 4529  {csn 4555  ⟨cop 4561  ∪ cuni 4838  ∪ ciun 4921  Disj wdisj 5039   class class class wbr 5072   ↦ cmpt 5153   Or wor 5525   × cxp 5616  ran crn 5619   “ cima 5621   ∘ ccom 5622   Fn wfn 6480  ⟶wf 6481  –1-1→wf1 6482  –onto→wfo 6483  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  Fincfn 8883  supcsup 9343  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171   − cmin 11368   / cdiv 11798  ℕcn 12165  2c2 12227  ℕ₀cn0 12428  ℤcz 12515  ℤ_≥cuz 12779  (,)cioo 13289  [,]cicc 13292  ...cfz 13452  ..^cfzo 13599  seqcseq 13954  ↑cexp 14014  abscabs 15187  topGenctg 17391  Topctop 22876  Clsdccld 22999  vol*covol 25447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-disj 5040  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-omul 8400  df-er 8633  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9314  df-sup 9345  df-inf 9346  df-oi 9415  df-dju 9816  df-card 9854  df-acn 9857  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-ioo 13293  df-ico 13295  df-icc 13296  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15441  df-rlim 15442  df-sum 15640  df-rest 17376  df-topgen 17397  df-psmet 21339  df-xmet 21340  df-met 21341  df-bl 21342  df-mopn 21343  df-top 22877  df-topon 22894  df-bases 22929  df-cld 23002  df-cmp 23370  df-conn 23395  df-ovol 25449  df-vol 25450

This theorem is referenced by:  mblfinlem4  38027  ismblfin  38028