Theorem mblfinlem2 38685

Description: Lemma for ismblfin 38688, 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 25089	. . . 4 ⊢ (topGen‘ran (,)) ∈ Top
2		0cld 23365	. . . 4 ⊢ ((topGen‘ran (,)) ∈ Top → ∅ ∈ (Clsd‘(topGen‘ran (,))))
3	1, 2	ax-mp 5	. . 3 ⊢ ∅ ∈ (Clsd‘(topGen‘ran (,)))
4		simpl3 1212	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘𝐴))
5		fveq2 6890	. . . . . 6 ⊢ (𝐴 = ∅ → (vol*‘𝐴) = (vol*‘∅))
6	5	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (vol*‘𝐴) = (vol*‘∅))
7	4, 6	breqtrd 5130	. . . 4 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘∅))
8		0ss 4349	. . . 4 ⊢ ∅ ⊆ 𝐴
9	7, 8	jctil 529	. . 3 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅)))
10		sseq1 3934	. . . . 5 ⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
11		fveq2 6890	. . . . . 6 ⊢ (𝑠 = ∅ → (vol*‘𝑠) = (vol*‘∅))
12	11	breq2d 5114	. . . . 5 ⊢ (𝑠 = ∅ → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∅)))
13	10, 12	anbi12d 644	. . . 4 ⊢ (𝑠 = ∅ → ((𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)) ↔ (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅))))
14	13	rspcev 3576	. . 3 ⊢ ((∅ ∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
15	3, 9, 14	sylancr 599	. 2 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 = ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
16		mblfinlem1 38684	. . . 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 1204	. . 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 1212	. . . . . . . . 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 6837	. . . . . . . . . . . . . . 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 6258	. . . . . . . . . . . . . . . . 17 ⊢ ran ([,] ∘ 𝑓) = ([,] “ ran 𝑓)
21		forn 6804	. . . . . . . . . . . . . . . . . 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 6054	. . . . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . . . 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 4879	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦)))
28		oveq1 7431	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1))
29	28	oveq1d 7439	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦)))
30	27, 29	opeq12d 4847	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩)
31		oveq2 7432	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣))
32	31	oveq2d 7440	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣)))
33	31	oveq2d 7440	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣)))
34	32, 33	opeq12d 4847	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))⟩)
35	30, 34	cbvmpov 7519	. . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧))
37	36	sseq1d 3940	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐)))
38		eqeq1 2764	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑧 → (𝑎 = 𝑐 ↔ 𝑧 = 𝑐))
39	37, 38	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
40	39	ralbidv 3185	. . . . . . . . . . . . . . . 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 3422	. . . . . . . . . . . . . . 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 4004	. . . . . . . . . . . . . . . 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 25929	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 2798	. . . . . . . . . . . 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 38683	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
48	47	3ad2ant1 1151	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 6894	. . . . . . . . . 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 6829	. . . . . . . . . . . . 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 4004	. . . . . . . . . . . . . 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 25921	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶( ≤ ∩ (ℝ × ℝ))
55		frn 6722	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 14 ⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ( ≤ ∩ (ℝ × ℝ))
58	53, 57	sstri 3918	. . . . . . . . . . . . 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 6731	. . . . . . . . . . . . 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 598	. . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . . . . 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 7086	. . . . . . . . . . . . . . . . . . . 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 3907	. . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 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 7086	. . . . . . . . . . . . . . . . . . . 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 3907	. . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . 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 25926	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 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 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)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
71		df-3or 1104	. . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . 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 3640	. . . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑚) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑚)))
75	74	sseq1d 3940	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑚) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐)))
76		eqeq1 2764	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑚) → (𝑎 = 𝑐 ↔ (𝑓‘𝑚) = 𝑐))
77	75, 76	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝑓‘𝑚) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)))
78	77	ralbidv 3185	. . . . . . . . . . . . . . . . . . . . . . . 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 3644	. . . . . . . . . . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑧) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑧)))
82	81	sseq2d 3941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑧) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧))))
83		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑧) → ((𝑓‘𝑚) = 𝑐 ↔ (𝑓‘𝑚) = (𝑓‘𝑧)))
84	82, 83	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (𝑓‘𝑧) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧))))
85	84	rspcva 3574	. . . . . . . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . . . . . . 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 3640	. . . . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = (𝑓‘𝑧) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑧)))
89	88	sseq1d 3940	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑧) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐)))
90		eqeq1 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = (𝑓‘𝑧) → (𝑎 = 𝑐 ↔ (𝑓‘𝑧) = 𝑐))
91	89, 90	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = (𝑓‘𝑧) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)))
92	91	ralbidv 3185	. . . . . . . . . . . . . . . . . . . . . . . . 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 3644	. . . . . . . . . . . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑐 = (𝑓‘𝑚) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑚)))
96	95	sseq2d 3941	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑚) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))))
97		eqeq2 2772	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = (𝑓‘𝑚) → ((𝑓‘𝑧) = 𝑐 ↔ (𝑓‘𝑧) = (𝑓‘𝑚)))
98	96, 97	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = (𝑓‘𝑚) → ((([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚))))
99	98	rspcva 3574	. . . . . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑧) = (𝑓‘𝑚) ↔ (𝑓‘𝑚) = (𝑓‘𝑧))
102	100, 101	imbitrdi 254	. . . . . . . . . . . . . . . . . . . . 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 873	. . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . . . 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 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)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧)))
107		f1of1 6828	. . . . . . . . . . . . . . . . . 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 7264	. . . . . . . . . . . . . . . . . 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 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)) → ((𝑓‘𝑚) = (𝑓‘𝑧) → 𝑚 = 𝑧))
110	106, 109	syld 48	. . . . . . . . . . . . . . . 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 981	. . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . 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 3205	. . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → (𝑚 = 𝑝 ↔ 𝑧 = 𝑝))
115		2fveq3 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑧 → ((,)‘(𝑓‘𝑚)) = ((,)‘(𝑓‘𝑧)))
116	115	ineq1d 4115	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))))
117	116	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅ ↔ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
118	114, 117	orbi12d 932	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑧 → ((𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
119	118	ralbidv 3185	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑧 → (∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
120	119	cbvralvw 3240	. . . . . . . . . . . . . 14 ⊢ (∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
121		eqeq2 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑝 → (𝑚 = 𝑧 ↔ 𝑚 = 𝑝))
122		2fveq3 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑝 → ((,)‘(𝑓‘𝑧)) = ((,)‘(𝑓‘𝑝)))
123	122	ineq2d 4116	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑝 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))))
124	123	eqeq1d 2762	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑝 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
125	121, 124	orbi12d 932	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑝 → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)))
126	125	cbvralvw 3240	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
127	126	ralbii 3108	. . . . . . . . . . . . . 14 ⊢ (∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
128	122	disjor 5084	. . . . . . . . . . . . . 14 ⊢ (Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))
129	120, 127, 128	3bitr4ri 307	. . . . . . . . . . . . 13 ⊢ (Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))
130	113, 129	sylibr 237	. . . . . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
132	60, 130, 131	uniiccvol 25910	. . . . . . . . . . 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 487	. . . . . . . . . 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 2797	. . . . . . . . 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 5130	. . . . . . . 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 15514	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
137		subf 11566	. . . . . . . . . . . 12 ⊢ − :(ℂ × ℂ)⟶ℂ
138		fco 6739	. . . . . . . . . . . 12 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ)
139	136, 137, 138	mp2an 705	. . . . . . . . . . 11 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ
140		zre 12704	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
141		2re 12424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℝ
142		reexpcl 14228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℝ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℝ)
143	141, 142	mpan 703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℝ)
144		2cn 12425	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℂ
145		2ne0 12456	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
146		nn0z 12724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ)
147		expne0i 14244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0)
148	144, 145, 146, 147	mp3an12i 1494	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → (2↑𝑦) ≠ 0)
149	143, 148	jca 521	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0))
150		redivcl 12043	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → (𝑥 / (2↑𝑦)) ∈ ℝ)
151		peano2re 11490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
152		redivcl 12043	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
153	151, 152	syl3an1 1181	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
154	150, 153	opelxpd 5690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
155	154	3expb 1138	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0)) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
156	140, 149, 155	syl2an 608	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
157	156	rgen2 3202	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ)
158		eqid 2760	. . . . . . . . . . . . . . . . . 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 8083	. . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ)
161		frn 6722	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 14 ⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ × ℝ)
164	53, 163	sstri 3918	. . . . . . . . . . . . 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 11264	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℂ
166		xpss12 5666	. . . . . . . . . . . . . 14 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ) ⊆ (ℂ × ℂ))
167	165, 165, 166	mp2an 705	. . . . . . . . . . . . 13 ⊢ (ℝ × ℝ) ⊆ (ℂ × ℂ)
168	164, 167	sstri 3918	. . . . . . . . . . . 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 6731	. . . . . . . . . . . 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 704	. . . . . . . . . . 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 6739	. . . . . . . . . . 11 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ × ℂ)) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ)
172	139, 170, 171	sylancr 599	. . . . . . . . . 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 13011	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
174		1z 12733	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
175	174	a1i 11	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → 1 ∈ ℤ)
176		ffvelcdm 7086	. . . . . . . . . . 11 ⊢ ((((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ ∧ 𝑛 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑛) ∈ ℝ)
177	173, 175, 176	serfre 14181	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ)
178		frn 6722	. . . . . . . . . . 11 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ)
179		ressxr 11360	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℝ*
180	178, 179	sstrdi 3921	. . . . . . . . . 10 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
181	52, 172, 177, 180	4syl 20	. . . . . . . . 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 11362	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℝ → 𝑀 ∈ ℝ*)
183	182	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → 𝑀 ∈ ℝ*)
184		supxrlub 13462	. . . . . . . . 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 609	. . . . . . . 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 235	. . . . . . 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 14163	. . . . . . . . . 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 6642	. . . . . . . . 9 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ↔ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn (ℤ≥‘1))
190	188, 189	mpbir 234	. . . . . . . 8 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn ℕ
191		breq2 5106	. . . . . . . . 9 ⊢ (𝑧 = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) → (𝑀 < 𝑧 ↔ 𝑀 < (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)))
192	191	rexrn 7092	. . . . . . . 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 221	. . . . . 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 7089	. . . . . . . . . . . . . . . 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 12451	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ≤ 0
197		df-br 5103	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ≤ 0 ↔ ⟨0, 0⟩ ∈ ≤ )
198	196, 197	mpbi 233	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ ≤
199		0re 11317	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
200		opelxpi 5688	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
201	199, 199, 200	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
202		elin 4092	. . . . . . . . . . . . . . . . 17 ⊢ (⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (⟨0, 0⟩ ∈ ≤ ∧ ⟨0, 0⟩ ∈ (ℝ × ℝ)))
203	198, 201, 202	mpbir2an 724	. . . . . . . . . . . . . . . 16 ⊢ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))
204		ifcl 4527	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)) ∧ ⟨0, 0⟩ ∈ ( ≤ ∩ (ℝ × ℝ))) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ ( ≤ ∩ (ℝ × ℝ)))
205	195, 203, 204	sylancl 598	. . . . . . . . . . . . . . 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 7120	. . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0(,)0) = ((,)‘⟨0, 0⟩)
208		iooid 13511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0(,)0) = ∅
209	207, 208	eqtr3i 2785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((,)‘⟨0, 0⟩) = ∅
210	209	ineq1i 4112	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = (∅ ∩ ((,)‘(𝑓‘𝑧)))
211		0in 4346	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅ ∩ ((,)‘(𝑓‘𝑧))) = ∅
212	210, 211	eqtri 2783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅
213	212	olci 880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 ∨ (((,)‘⟨0, 0⟩) ∩ ((,)‘(𝑓‘𝑧))) = ∅)
214		ineq1 4109	. . . . . . . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . . . . 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 929	. . . . . . . . . . . . . . . . . . 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 4109	. . . . . . . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . . . . 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 929	. . . . . . . . . . . . . . . . . . 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 4521	. . . . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . . . 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 4113	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅)
223		in0 4344	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ∅) = ∅
224	222, 223	eqtri 2783	. . . . . . . . . . . . . . . . . 18 ⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅
225	224	olci 880	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) ∩ ((,)‘⟨0, 0⟩)) = ∅)
226		ineq2 4110	. . . . . . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . . . 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 929	. . . . . . . . . . . . . . . . . 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 4110	. . . . . . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . . . 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 929	. . . . . . . . . . . . . . . . . 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 4521	. . . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . . 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 3205	. . . . . . . . . . . . . . 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 5073	. . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑚 → (𝑧 ∈ (1...𝑛) ↔ 𝑚 ∈ (1...𝑛)))
237		fveq2 6890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑚 → (𝑓‘𝑧) = (𝑓‘𝑚))
238	236, 237	ifbieq1d 4506	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑚 → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩))
239		eqid 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))
240		fvex 6903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓‘𝑚) ∈ V
241		opex 5432	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨0, 0⟩ ∈ V
242	240, 241	ifex 4532	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩) ∈ V
243	238, 239, 242	fvmpt 6998	. . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ → ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚)) = ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩)))
245		fvif 6906	. . . . . . . . . . . . . . . . . 18 ⊢ ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩))
246	244, 245	eqtrdi 2811	. . . . . . . . . . . . . . . . 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 3082	. . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑧 → (𝑚 ∈ (1...𝑛) ↔ 𝑧 ∈ (1...𝑛)))
249	248, 115	ifbieq1d 4506	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑧 → if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘⟨0, 0⟩)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘⟨0, 0⟩)))
250	249	disjor 5084	. . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . 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 25910	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 11361	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
257	164, 256	sstri 3918	. . . . . . . . . . . . . . . . . . . 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 3907	. . . . . . . . . . . . . . . . . . 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 11363	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ∈ ℝ*
260		opelxpi 5688	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ⟨0, 0⟩ ∈ (ℝ* × ℝ*))
261	259, 259, 260	mp2an 705	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨0, 0⟩ ∈ (ℝ* × ℝ*)
262		ifcl 4527	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑧) ∈ (ℝ* × ℝ*) ∧ ⟨0, 0⟩ ∈ (ℝ* × ℝ*)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ (ℝ* × ℝ*))
263	258, 261, 262	sylancl 598	. . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . 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 13586	. . . . . . . . . . . . . . . . . . . 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 6958	. . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . 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 7135	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 5921	. . . . . . . . . . . . . . 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 4879	. . . . . . . . . . . . . 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 12354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
274	273, 173	eleqtrdi 2870	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ (ℤ≥‘1))
275		fzouzsplit 13836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘1) → (ℤ≥‘1) = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
276	274, 275	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (ℤ≥‘1) = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
277	173, 276	eqtrid 2807	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ℕ = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
278		nnz 12721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
279		fzval3 13876	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℤ → (1...𝑛) = (1..^(𝑛 + 1)))
280	278, 279	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (1...𝑛) = (1..^(𝑛 + 1)))
281	280	uneq1d 4165	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1))) = ((1..^(𝑛 + 1)) ∪ (ℤ≥‘(𝑛 + 1))))
282	277, 281	eqtr4d 2798	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ℕ = ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1))))
283		fvif 6906	. . . . . . . . . . . . . . . . . 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 4979	. . . . . . . . . . . . . . . 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 6903	. . . . . . . . . . . . . . . . 17 ⊢ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) ∈ V
287	286	dfiun3 5953	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) = ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))
288		iunxun 5053	. . . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . 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 4487	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘(𝑓‘𝑧)))
291	290	iuneq2i 4972	. . . . . . . . . . . . . . . . 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 13751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (ℤ≥‘(𝑛 + 1)) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
294	293	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)))
295		nncn 12350	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
296		ax-1cn 11265	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℂ
297		pncan 11570	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
298	295, 296, 297	sylancl 598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛)
299	298	oveq2d 7440	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ → (1...((𝑛 + 1) − 1)) = (1...𝑛))
300	299	eleq2d 2846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℕ → (𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ 𝑧 ∈ (1...𝑛)))
301	300	notbid 321	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
302	301	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛)))
303	294, 302	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...𝑛))
304	303	iffalsed 4492	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘⟨0, 0⟩)) = ([,]‘⟨0, 0⟩))
305	304	iuneq2dv 4975	. . . . . . . . . . . . . . . 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 4167	. . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . 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 6894	. . . . . . . . . . . 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 13277	. . . . . . . . . . . . . . 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 13012	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
313	312	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ (ℤ≥‘1))
314	313	adantl 487	. . . . . . . . . . . . . . . 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 13694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (1...𝑛) → 𝑢 ∈ ℕ)
316	172	ffvelcdmda 7089	. . . . . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ)
320	319	adantl 487	. . . . . . . . . . . . . . . 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 14172	. . . . . . . . . . . . . . 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 11366	. . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . 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 4487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ (1...𝑛) → if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), ⟨0, 0⟩) = (𝑓‘𝑚))
325	238, 324	sylan9eqr 2817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ (1...𝑛) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = (𝑓‘𝑚))
326		elfznn 13694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (1...𝑛) → 𝑚 ∈ ℕ)
327	240	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (1...𝑛) → (𝑓‘𝑚) ∈ V)
328	323, 325, 326, 327	fvmptd 7006	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ (1...𝑛) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = (𝑓‘𝑚))
329	328	adantl 487	. . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . 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 6903	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓‘𝑧) ∈ V
332	331, 241	ifex 4532	. . . . . . . . . . . . . . . . . . . . 21 ⊢ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ V
333	332, 239	fnmpti 6687	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)) Fn ℕ
334		fvco2 6987	. . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . 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 6714	. . . . . . . . . . . . . . . . . . 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 6987	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs ∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚)))
339	337, 326, 338	syl2an 608	. . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 14176	. . . . . . . . . . . . . . 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 3907	. . . . . . . . . . . . . . . . . . . . . 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 11305	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℂ
346		opelxpi 5688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ) → ⟨0, 0⟩ ∈ (ℂ × ℂ))
347	345, 345, 346	mp2an 705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ⟨0, 0⟩ ∈ (ℂ × ℂ)
348		ifcl 4527	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓‘𝑧) ∈ (ℂ × ℂ) ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) ∈ (ℂ × ℂ))
349	344, 347, 348	sylancl 598	. . . . . . . . . . . . . . . . . . . . 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 7120	. . . . . . . . . . . . . . . . . . . 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 6739	. . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . 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 7089	. . . . . . . . . . . . . . . . . 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 14181	. . . . . . . . . . . . . . . . 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 6715	. . . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ran seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩)))))
359	354	frnd 6723	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 3909	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑚 + 𝑢) ∈ ℝ)
364	363	adantl 487	. . . . . . . . . . . . . . . . . . . . . 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 11297	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℝ → 𝑚 ∈ ℂ)
366		recn 11297	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 ∈ ℝ → 𝑢 ∈ ℂ)
367		recn 11297	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 ∈ ℝ → 𝑣 ∈ ℂ)
368		addass 11294	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
369	365, 366, 367, 368	syl3an 1178	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣)))
370	369	adantl 487	. . . . . . . . . . . . . . . . . . . . . 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 12762	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 < 𝑡 ↔ (𝑛 + 1) ≤ 𝑡))
372	371	biimpa 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑛 + 1) ≤ 𝑡)
373	273	nnzd 12726	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℤ)
374		nnz 12721	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ ℤ)
375		eluz 12986	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑛 + 1) ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
376	373, 374, 375	syl2an 608	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
377	376	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡))
378	372, 377	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1)))
379	378	adantlll 731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1)))
380	313	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . . . 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 13694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℕ)
383	381, 382, 353	syl2an 608	. . . . . . . . . . . . . . . . . . . . . 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 14185	. . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . . 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 13663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℤ)
387	386	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
388		0red 11318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
389	273	nnred 12357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
390	389	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
391	386	zred 12810	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℝ)
392	391	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
393	273	nngt0d 12394	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 ∈ ℕ → 0 < (𝑛 + 1))
394	393	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
395		elfzle1 13667	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → (𝑛 + 1) ≤ 𝑚)
396	395	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
397	388, 390, 392, 394, 396	ltletrd 11477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
398		elnnz 12710	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ ℕ ↔ (𝑚 ∈ ℤ ∧ 0 < 𝑚))
399	387, 397, 398	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ)
400	333, 399, 334	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12349	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
403	402	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 ∈ ℝ)
404	389	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ)
405	391	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ)
406	402	ltp1d 12254	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
407	406	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < (𝑛 + 1))
408	395	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚)
409	403, 404, 405, 407, 408	ltletrd 11477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < 𝑚)
410	409	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → 𝑛 < 𝑚)
411	403, 405	ltnled 11464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛))
412		breq1 5105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑚 = 𝑧 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛))
413	412	equcoms 2052	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑧 = 𝑚 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛))
414	413	notbid 321	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 = 𝑚 → (¬ 𝑚 ≤ 𝑛 ↔ ¬ 𝑧 ≤ 𝑛))
415	411, 414	sylan9bb 519	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → (𝑛 < 𝑚 ↔ ¬ 𝑧 ≤ 𝑛))
416	410, 415	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ≤ 𝑛)
417		elfzle2 13668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ≤ 𝑛)
418	416, 417	nsyl 141	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ∈ (1...𝑛))
419	418	iffalsed 4492	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = ⟨0, 0⟩)
420	386	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ)
421		0red 11318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ)
422	393	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1))
423	421, 404, 405, 422, 408	ltletrd 11477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚)
424	420, 423, 398	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩))‘𝑚) = ⟨0, 0⟩)
427	426	ad4ant14 765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6990	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ ⟨0, 0⟩ ∈ (ℂ × ℂ)) → ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩)))
431	137, 347, 430	mp2an 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((abs ∘ − )‘⟨0, 0⟩) = (abs‘( − ‘⟨0, 0⟩))
432		df-ov 7427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 − 0) = ( − ‘⟨0, 0⟩)
433		0m0e0 12468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 − 0) = 0
434	432, 433	eqtr3i 2785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ( − ‘⟨0, 0⟩) = 0
435	434	fveq2i 6893	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (abs‘( − ‘⟨0, 0⟩)) = (abs‘0)
436		abs0 15461	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (abs‘0) = 0
437	435, 436	eqtri 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (abs‘( − ‘⟨0, 0⟩)) = 0
438	431, 437	eqtri 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((abs ∘ − )‘⟨0, 0⟩) = 0
439	429, 438	eqtrdi 2811	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ (ℤ≥‘(𝑛 + 1)))
441		c0ex 11307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ V
442	441	fvconst2 7214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑚 ∈ (ℤ≥‘(𝑛 + 1)) → (((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0)
443	440, 442	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → (((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0)
444	443	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0)
445	439, 444	eqtr4d 2798	. . . . . . . . . . . . . . . . . . . . . . . . . 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 14176	. . . . . . . . . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1))
448	447	ser0 14204	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) → (seq(𝑛 + 1)( + , ((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡) = 0)
449	378, 448	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡) = 0)
450	446, 449	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . . 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 7442	. . . . . . . . . . . . . . . . . . . . . 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 7089	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑚 + 𝑣) ∈ ℝ)
457	456	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 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 14172	. . . . . . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . . . 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 11344	. . . . . . . . . . . . . . . . . . . . . . 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 11517	. . . . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . . . . . . . . . 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 14172	. . . . . . . . . . . . . . . . . . . . 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 11889	. . . . . . . . . . . . . . . . . . . 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 5126	. . . . . . . . . . . . . . . . . . 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 13012	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 ∈ ℕ ↔ 𝑡 ∈ (ℤ≥‘1))
470	469	biimpi 219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ (ℤ≥‘1))
471	470	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 13667	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ (1...𝑡) → 1 ≤ 𝑚)
475	474	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 1 ≤ 𝑚)
476	382	nnred 12357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℝ)
477	476	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℝ)
478		nnre 12349	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 ∈ ℕ → 𝑡 ∈ ℝ)
479	478	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ∈ ℝ)
480	402	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑛 ∈ ℝ)
481		elfzle2 13668	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ≤ 𝑡)
482	481	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑡)
483		simplr 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ≤ 𝑛)
484	477, 479, 480, 482, 483	letrd 11474	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑛)
485		elfzelz 13663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℤ)
486	278	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ ℤ)
487		elfz 13652	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑚 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
488	174, 487	mp3an2 1478	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
489	485, 486, 488	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛)))
490	475, 484, 489	mpbir2and 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛))
491	490	ad5ant2345 1397	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 ∈ (1...𝑛))
494		iftrue 4487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), ⟨0, 0⟩) = (𝑓‘𝑧))
495	493, 494	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . 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 7006	. . . . . . . . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . . . 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 6990	. . . . . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . . . . . . 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 14176	. . . . . . . . . . . . . . . . . . . 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 12986	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ (ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛))
510	374, 278, 509	syl2anr 609	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 ∈ (ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛))
511	510	biimpar 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ (ℤ≥‘𝑡))
512	511	adantlll 731	. . . . . . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . . . . . . 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 13663	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℤ)
515	514	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℤ)
516		0red 11318	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ∈ ℝ)
517		peano2nn 12354	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℕ)
518	517	nnred 12357	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈ ℝ)
519	518	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ∈ ℝ)
520	514	zred 12810	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℝ)
521	520	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℝ)
522	517	nngt0d 12394	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 ∈ ℕ → 0 < (𝑡 + 1))
523	522	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < (𝑡 + 1))
524		elfzle1 13667	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → (𝑡 + 1) ≤ 𝑚)
525	524	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ≤ 𝑚)
526	516, 519, 521, 523, 525	ltletrd 11477	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < 𝑚)
527	515, 526, 398	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
528	527	adantlr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑡 ∈ ℕ ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
529	528	adantlll 731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ)
530	170	ffvelcdmda 7089	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7086	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) → ( − ‘(𝑓‘𝑚)) ∈ ℂ)
532	137, 530, 531	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . 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 15623	. . . . . . . . . . . . . . . . . . . . . . . 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 6990	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (( − :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) → ((abs ∘ − )‘(𝑓‘𝑚)) = (abs‘( − ‘(𝑓‘𝑚))))
535	137, 530, 534	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . . . . . . . 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 5132	. . . . . . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . . . . . . . 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 14184	. . . . . . . . . . . . . . . . . . . 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 5126	. . . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . 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 11425	. . . . . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . . . . . . . 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 7093	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . 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 3254	. . . . . . . . . . . . . . 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 11468	. . . . . . . . . . . . . 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 9467	. . . . . . . . . . . . 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 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)} ∧ 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 2804	. . . . . . . . . . 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 13694	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ∈ ℕ)
557	164, 65	sselid 3907	. . . . . . . . . . . . . . . . 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 8044	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (𝑓‘𝑧) = ⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩)
559	558	fveq2d 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) = ([,]‘⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩))
560		df-ov 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) = ([,]‘⟨(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))⟩)
561	559, 560	eqtr4di 2813	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) = ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))))
562		xp1st 8037	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (1st ‘(𝑓‘𝑧)) ∈ ℝ)
563		xp2nd 8038	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → (2nd ‘(𝑓‘𝑧)) ∈ ℝ)
564		iccssre 13567	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑧)) ∈ ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ)
565	562, 563, 564	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ)
566	561, 565	eqsstrd 3943	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) ⊆ ℝ)
567	557, 566	syl 18	. . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . . 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 5002	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ)
571	569, 570	sylibr 237	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 12987	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ ℤ → (𝑛 + 1) ∈ (ℤ≥‘(𝑛 + 1)))
574		ne0i 4286	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ (ℤ≥‘(𝑛 + 1)) → (ℤ≥‘(𝑛 + 1)) ≠ ∅)
575		iunconst 4960	. . . . . . . . . . . . . . . 16 ⊢ ((ℤ≥‘(𝑛 + 1)) ≠ ∅ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) = ([,]‘⟨0, 0⟩))
576	373, 573, 574, 575	4syl 20	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) = ([,]‘⟨0, 0⟩))
577		iccid 13528	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ* → (0[,]0) = {0})
578	259, 577	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (0[,]0) = {0}
579		df-ov 7427	. . . . . . . . . . . . . . . 16 ⊢ (0[,]0) = ([,]‘⟨0, 0⟩)
580	578, 579	eqtr3i 2785	. . . . . . . . . . . . . . 15 ⊢ {0} = ([,]‘⟨0, 0⟩)
581	576, 580	eqtr4di 2813	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) = {0})
582		snssi 4742	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
583	199, 582	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ {0} ⊆ ℝ
584	581, 583	eqsstrdi 3953	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩) ⊆ ℝ)
585	584	adantl 487	. . . . . . . . . . . 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 6894	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘⟨0, 0⟩)) = (vol*‘{0}))
587	586	adantl 487	. . . . . . . . . . . . 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 25825	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ → (vol*‘{0}) = 0)
589	199, 588	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol*‘{0}) = 0
590	587, 589	eqtrdi 2811	. . . . . . . . . . . 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 25830	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 2795	. . . . . . . . . 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 5114	. . . . . . . . 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 232	. . . . . . . 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 3175	. . . . . . 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 487	. . . . . 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 16	. . . . 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 14122	. . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin
600		icccld 25094	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd ‘(𝑓‘𝑧)) ∈ ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
601	562, 563, 600	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran (,))))
602	561, 601	eqeltrd 2860	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) → ([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
603	557, 602	syl 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)) ∈ (Clsd‘(topGen‘ran (,))))
604	556, 603	sylan2 605	. . . . . . . . . . 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 3154	. . . . . . . . . 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 25090	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
607	606	iuncld 23372	. . . . . . . . . 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 1494	. . . . . . . . 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 486	. . . . . . . 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 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑓‘𝑧) → ([,]‘𝑏) = ([,]‘(𝑓‘𝑧)))
611	610	sseq1d 3940	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑓‘𝑧) → (([,]‘𝑏) ⊆ 𝐴 ↔ ([,]‘(𝑓‘𝑧)) ⊆ 𝐴))
612	611	elrab 3644	. . . . . . . . . . . . . 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 503	. . . . . . . . . . . . 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 19	. . . . . . . . . . . 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 605	. . . . . . . . . . 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 3154	. . . . . . . . . 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 5002	. . . . . . . . . 10 ⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴)
618	616, 617	sylibr 237	. . . . . . . . 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 486	. . . . . . . 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 785	. . . . . . . 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 3934	. . . . . . . . . 10 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑠 ⊆ 𝐴 ↔ ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴))
622		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (vol*‘𝑠) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))
623	622	breq2d 5114	. . . . . . . . . 10 ⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))
624	621, 623	anbi12d 644	. . . . . . . . 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 3576	. . . . . . . 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 850	. . . . . . 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 592	. . . . . 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 727	. . . . 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 3169	. . . 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 728	. . 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 1967	. 2 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) ∧ 𝐴 ≠ ∅) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
632	15, 631	pm2.61dane 3042	1 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   ∪ cun 4148  ∅c0 4277  ifcif 4481  𝒫 cpw 4556  {csn 4584  ⟨cop 4834  ∪ cuni 4866  ∪ ciun 4950  Disj wdisj 5069   class class class wbr 5102   ↦ cmpt 5185   Or wor 5558   × cxp 5649  ran crn 5652   “ cima 5654   ∘ ccom 5655   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  1^st c1st 8003  2^nd c2nd 8004  Fincfn 8980  supcsup 9439  ℂcc 11205  ℝcr 11206  0cc0 11207  1c1 11208   + caddc 11210  ℝ^*cxr 11349   < clt 11350   ≤ cle 11351   − cmin 11548   / cdiv 11980  ℕcn 12342  2c2 12404  ℕ₀cn0 12613  ℤcz 12700  ℤ_≥cuz 12972  (,)cioo 13483  [,]cicc 13486  ...cfz 13646  ..^cfzo 13795  seqcseq 14151  ↑cexp 14211  abscabs 15410  topGenctg 17617  Topctop 23220  Clsdccld 23343  vol*covol 25792

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7697  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-oadd 8487  df-omul 8488  df-er 8724  df-map 8856  df-pm 8857  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-fi 9410  df-sup 9441  df-inf 9442  df-oi 9511  df-dju 9989  df-card 10027  df-acn 10030  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-3 12413  df-n0 12614  df-z 12701  df-uz 12973  df-q 13083  df-rp 13128  df-xneg 13248  df-xadd 13249  df-xmul 13250  df-ioo 13487  df-ico 13489  df-icc 13490  df-fz 13647  df-fzo 13796  df-fl 13939  df-seq 14152  df-exp 14212  df-hash 14482  df-cj 15275  df-re 15276  df-im 15277  df-sqrt 15411  df-abs 15412  df-clim 15664  df-rlim 15665  df-sum 15863  df-rest 17602  df-topgen 17623  df-psmet 21679  df-xmet 21680  df-met 21681  df-bl 21682  df-mopn 21683  df-top 23221  df-topon 23238  df-bases 23273  df-cld 23346  df-cmp 23714  df-conn 23739  df-ovol 25794  df-vol 25795

This theorem is used by:  mblfinlem4  38687  ismblfin  38688