sxbrsigalem2 - Mathbox for Thierry Arnoux

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Theorem sxbrsigalem2 31776

Description: The sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed half-spaces of (ℝ × ℝ). The proof goes by noting the fact that the dyadic rectangles are intersections of a 'vertical band' and an 'horizontal band', which themselves are differences of closed half-spaces. (Contributed by Thierry Arnoux, 17-Sep-2017.)

**Hypotheses**
Ref	Expression
sxbrsiga.0	⊢ 𝐽 = (topGen‘ran (,))
dya2ioc.1	⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
dya2ioc.2	⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))

**Assertion**
Ref	Expression
sxbrsigalem2	⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))

Distinct variable groups: 𝑥,𝑛 𝑥,𝐼 𝑣,𝑢,𝐼,𝑥 𝑢,𝑛,𝑣 𝑅,𝑛,𝑥 𝑥,𝐽 𝑒,𝑓,𝑛,𝑢,𝑣,𝑥 Allowed substitution hints: 𝑅(𝑣,𝑢,𝑒,𝑓) 𝐼(𝑒,𝑓,𝑛) 𝐽(𝑣,𝑢,𝑒,𝑓,𝑛)

Proof of Theorem sxbrsigalem2 Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sxbrsiga.0	. . . 4 ⊢ 𝐽 = (topGen‘ran (,))
2		dya2ioc.1	. . . 4 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
3		dya2ioc.2	. . . 4 ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))
4	1, 2, 3	dya2iocucvr 31774	. . 3 ⊢ ∪ ran 𝑅 = (ℝ × ℝ)
5		sxbrsigalem0 31761	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
6	4, 5	eqtr4i 2784	. 2 ⊢ ∪ ran 𝑅 = ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
7		vex 3413	. . . . . 6 ⊢ 𝑢 ∈ V
8		vex 3413	. . . . . 6 ⊢ 𝑣 ∈ V
9	7, 8	xpex 7479	. . . . 5 ⊢ (𝑢 × 𝑣) ∈ V
10	3, 9	elrnmpo 7287	. . . 4 ⊢ (𝑑 ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼 𝑑 = (𝑢 × 𝑣))
11		simpr 488	. . . . . . 7 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑑 = (𝑢 × 𝑣)) → 𝑑 = (𝑢 × 𝑣))
12	1, 2	dya2icobrsiga 31766	. . . . . . . . . . . . 13 ⊢ ran 𝐼 ⊆ 𝔅_ℝ
13		brsigasspwrn 31676	. . . . . . . . . . . . 13 ⊢ 𝔅_ℝ ⊆ 𝒫 ℝ
14	12, 13	sstri 3903	. . . . . . . . . . . 12 ⊢ ran 𝐼 ⊆ 𝒫 ℝ
15	14	sseli 3890	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ∈ 𝒫 ℝ)
16	15	elpwid 4508	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ⊆ ℝ)
17	14	sseli 3890	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ∈ 𝒫 ℝ)
18	17	elpwid 4508	. . . . . . . . . 10 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ⊆ ℝ)
19		xpinpreima2 31382	. . . . . . . . . 10 ⊢ ((𝑢 ⊆ ℝ ∧ 𝑣 ⊆ ℝ) → (𝑢 × 𝑣) = ((^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∩ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣)))
20	16, 18, 19	syl2an 598	. . . . . . . . 9 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 × 𝑣) = ((^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∩ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣)))
21		reex 10671	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
22	21	mptex 6982	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
23	22	rnex 7627	. . . . . . . . . . . . . . 15 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
24	21	mptex 6982	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
25	24	rnex 7627	. . . . . . . . . . . . . . 15 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
26	23, 25	unex 7472	. . . . . . . . . . . . . 14 ⊢ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ∈ V
27	26	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ∈ V)
28	27	sgsiga 31633	. . . . . . . . . . . 12 ⊢ (⊤ → (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra)
29	28	mptru 1545	. . . . . . . . . . 11 ⊢ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra
30	29	a1i 11	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra)
31		1stpreima 30567	. . . . . . . . . . . . 13 ⊢ (𝑢 ⊆ ℝ → (^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) = (𝑢 × ℝ))
32	16, 31	syl 17	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ran 𝐼 → (^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) = (𝑢 × ℝ))
33		ovex 7188	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V
34	2, 33	elrnmpo 7287	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ran 𝐼 ↔ ∃𝑥 ∈ ℤ ∃𝑛 ∈ ℤ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
35		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
36	35	xpeq1d 5556	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (𝑢 × ℝ) = (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ))
37		difxp1 5998	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∖ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
38		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℤ)
39	38	zred 12131	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
40		2rp 12440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ⁺
41	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 2 ∈ ℝ⁺)
42		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
43	41, 42	rpexpcld 13663	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2↑𝑛) ∈ ℝ⁺)
44	39, 43	rerpdivcld 12508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ)
45	44	rexrd 10734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ^*)
46		1red 10685	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 1 ∈ ℝ)
47	39, 46	readdcld 10713	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 + 1) ∈ ℝ)
48	47, 43	rerpdivcld 12508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ)
49	48	rexrd 10734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ^*)
50		pnfxr 10738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ +∞ ∈ ℝ^*
51	50	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → +∞ ∈ ℝ^*)
52	39	lep1d 11614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ≤ (𝑥 + 1))
53	39, 47, 43, 52	lediv1dd 12535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ≤ ((𝑥 + 1) / (2↑𝑛)))
54		pnfge 12571	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 + 1) / (2↑𝑛)) ∈ ℝ^* → ((𝑥 + 1) / (2↑𝑛)) ≤ +∞)
55	49, 54	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ≤ +∞)
56		difico 30632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑥 / (2↑𝑛)) ∈ ℝ^* ∧ ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ ((𝑥 / (2↑𝑛)) ≤ ((𝑥 + 1) / (2↑𝑛)) ∧ ((𝑥 + 1) / (2↑𝑛)) ≤ +∞)) → (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
57	45, 49, 51, 53, 55, 56	syl32anc 1375	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
58	57	xpeq1d 5556	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ))
59	37, 58	syl5reqr 2808	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ) = ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∖ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)))
60	29	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra)
61		ssun1 4079	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
62		eqid 2758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)
63		oveq1 7162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → (𝑒[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
64	63	xpeq1d 5556	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ))
65	64	rspceeqv 3558	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛)) ∈ ℝ ∧ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)) → ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
66	44, 62, 65	sylancl 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
67		eqid 2758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
68		ovex 7188	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒[,)+∞) ∈ V
69	68, 21	xpex 7479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
70	67, 69	elrnmpti 5805	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ↔ ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
71	66, 70	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
72	61, 71	sseldi 3892	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
73		elsigagen 31638	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ∈ V ∧ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
74	26, 72, 73	sylancr 590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
75		eqid 2758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)
76		oveq1 7162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → (𝑒[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
77	76	xpeq1d 5556	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
78	77	rspceeqv 3558	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 + 1) / (2↑𝑛)) ∈ ℝ ∧ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)) → ∃𝑒 ∈ ℝ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
79	48, 75, 78	sylancl 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑒 ∈ ℝ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
80	67, 69	elrnmpti 5805	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ↔ ∃𝑒 ∈ ℝ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
81	79, 80	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
82	61, 81	sseldi 3892	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
83		elsigagen 31638	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ∈ V ∧ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
84	26, 82, 83	sylancr 590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
85		difelsiga 31624	. . . . . . . . . . . . . . . . . . 19 ⊢ (((sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra ∧ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∧ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))) → ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∖ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
86	60, 74, 84, 85	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∖ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
87	59, 86	eqeltrd 2852	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
88	87	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
89	36, 88	eqeltrd 2852	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (𝑢 × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
90	89	ex 416	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) → (𝑢 × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))))
91	90	rexlimivv 3216	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℤ ∃𝑛 ∈ ℤ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) → (𝑢 × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
92	34, 91	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ran 𝐼 → (𝑢 × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
93	32, 92	eqeltrd 2852	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran 𝐼 → (^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
94	93	adantr 484	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
95		2ndpreima 30568	. . . . . . . . . . . . 13 ⊢ (𝑣 ⊆ ℝ → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) = (ℝ × 𝑣))
96	18, 95	syl 17	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran 𝐼 → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) = (ℝ × 𝑣))
97	2, 33	elrnmpo 7287	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran 𝐼 ↔ ∃𝑥 ∈ ℤ ∃𝑛 ∈ ℤ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
98		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
99	98	xpeq2d 5557	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (ℝ × 𝑣) = (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))))
100		difxp2 5999	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
101	57	xpeq2d 5557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))))
102	100, 101	syl5reqr 2808	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))))
103		ssun2 4080	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
104		eqid 2758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))
105		oveq1 7162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (𝑓[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
106	105	xpeq2d 5557	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)))
107	106	rspceeqv 3558	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛)) ∈ ℝ ∧ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))) → ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
108	44, 104, 107	sylancl 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
109		eqid 2758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
110		ovex 7188	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓[,)+∞) ∈ V
111	21, 110	xpex 7479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
112	109, 111	elrnmpti 5805	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ↔ ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
113	108, 112	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
114	103, 113	sseldi 3892	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
115		elsigagen 31638	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ∈ V ∧ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
116	26, 114, 115	sylancr 590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
117		eqid 2758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))
118		oveq1 7162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (𝑓[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
119	118	xpeq2d 5557	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
120	119	rspceeqv 3558	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 + 1) / (2↑𝑛)) ∈ ℝ ∧ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))) → ∃𝑓 ∈ ℝ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
121	48, 117, 120	sylancl 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑓 ∈ ℝ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
122	109, 111	elrnmpti 5805	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ↔ ∃𝑓 ∈ ℝ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
123	121, 122	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
124	103, 123	sseldi 3892	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
125		elsigagen 31638	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ∈ V ∧ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
126	26, 124, 125	sylancr 590	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
127		difelsiga 31624	. . . . . . . . . . . . . . . . . . 19 ⊢ (((sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra ∧ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∧ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))) → ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
128	60, 116, 126, 127	syl3anc 1368	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
129	102, 128	eqeltrd 2852	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
130	129	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
131	99, 130	eqeltrd 2852	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (ℝ × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
132	131	ex 416	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) → (ℝ × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))))
133	132	rexlimivv 3216	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℤ ∃𝑛 ∈ ℤ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) → (ℝ × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
134	97, 133	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran 𝐼 → (ℝ × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
135	96, 134	eqeltrd 2852	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐼 → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
136	135	adantl 485	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
137		inelsiga 31626	. . . . . . . . . 10 ⊢ (((sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra ∧ (^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∧ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))) → ((^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∩ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
138	30, 94, 136, 137	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → ((^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∩ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
139	20, 138	eqeltrd 2852	. . . . . . . 8 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
140	139	adantr 484	. . . . . . 7 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑑 = (𝑢 × 𝑣)) → (𝑢 × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
141	11, 140	eqeltrd 2852	. . . . . 6 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑑 = (𝑢 × 𝑣)) → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
142	141	ex 416	. . . . 5 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑑 = (𝑢 × 𝑣) → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))))
143	142	rexlimivv 3216	. . . 4 ⊢ (∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼 𝑑 = (𝑢 × 𝑣) → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
144	10, 143	sylbi 220	. . 3 ⊢ (𝑑 ∈ ran 𝑅 → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
145	144	ssriv 3898	. 2 ⊢ ran 𝑅 ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
146		sigagenss2 31641	. 2 ⊢ ((∪ ran 𝑅 = ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ∧ ran 𝑅 ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∧ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ∈ V) → (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
147	6, 145, 26, 146	mp3an 1458	1 ⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ∃wrex 3071 Vcvv 3409 ∖ cdif 3857 ∪ cun 3858 ∩ cin 3859 ⊆ wss 3860 𝒫 cpw 4497 ∪ cuni 4801 class class class wbr 5035 ↦ cmpt 5115 × cxp 5525 ^◡ccnv 5526 ran crn 5528 ↾ cres 5529 “ cima 5530 ‘cfv 6339 (class class class)co 7155 ∈ cmpo 7157 1^st c1st 7696 2^nd c2nd 7697 ℝcr 10579 1c1 10581 + caddc 10583 +∞cpnf 10715 ℝ^*cxr 10717 ≤ cle 10719 / cdiv 11340 2c2 11734 ℤcz 12025 ℝ⁺crp 12435 (,)cioo 12784 [,)cico 12786 ↑cexp 13484 topGenctg 16774 sigAlgebracsiga 31599 sigaGencsigagen 31629 𝔅_ℝcbrsiga 31672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-inf2 9142 ax-ac2 9928 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 ax-pre-sup 10658 ax-addf 10659 ax-mulf 10660

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-se 5487 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-2o 8118 df-er 8304 df-map 8423 df-pm 8424 df-ixp 8485 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-fi 8913 df-sup 8944 df-inf 8945 df-oi 9012 df-dju 9368 df-card 9406 df-acn 9409 df-ac 9581 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-div 11341 df-nn 11680 df-2 11742 df-3 11743 df-4 11744 df-5 11745 df-6 11746 df-7 11747 df-8 11748 df-9 11749 df-n0 11940 df-z 12026 df-dec 12143 df-uz 12288 df-q 12394 df-rp 12436 df-xneg 12553 df-xadd 12554 df-xmul 12555 df-ioo 12788 df-ioc 12789 df-ico 12790 df-icc 12791 df-fz 12945 df-fzo 13088 df-fl 13216 df-mod 13292 df-seq 13424 df-exp 13485 df-fac 13689 df-bc 13718 df-hash 13746 df-shft 14479 df-cj 14511 df-re 14512 df-im 14513 df-sqrt 14647 df-abs 14648 df-limsup 14881 df-clim 14898 df-rlim 14899 df-sum 15096 df-ef 15474 df-sin 15476 df-cos 15477 df-pi 15479 df-struct 16548 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-starv 16643 df-sca 16644 df-vsca 16645 df-ip 16646 df-tset 16647 df-ple 16648 df-ds 16650 df-unif 16651 df-hom 16652 df-cco 16653 df-rest 16759 df-topn 16760 df-0g 16778 df-gsum 16779 df-topgen 16780 df-pt 16781 df-prds 16784 df-xrs 16838 df-qtop 16843 df-imas 16844 df-xps 16846 df-mre 16920 df-mrc 16921 df-acs 16923 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-submnd 18028 df-mulg 18297 df-cntz 18519 df-cmn 18980 df-psmet 20163 df-xmet 20164 df-met 20165 df-bl 20166 df-mopn 20167 df-fbas 20168 df-fg 20169 df-cnfld 20172 df-refld 20375 df-top 21599 df-topon 21616 df-topsp 21638 df-bases 21651 df-cld 21724 df-ntr 21725 df-cls 21726 df-nei 21803 df-lp 21841 df-perf 21842 df-cn 21932 df-cnp 21933 df-haus 22020 df-cmp 22092 df-tx 22267 df-hmeo 22460 df-fil 22551 df-fm 22643 df-flim 22644 df-flf 22645 df-fcls 22646 df-xms 23027 df-ms 23028 df-tms 23029 df-cncf 23584 df-cfil 23960 df-cmet 23962 df-cms 24040 df-limc 24570 df-dv 24571 df-log 25252 df-cxp 25253 df-logb 25455 df-siga 31600 df-sigagen 31630 df-brsiga 31673

This theorem is referenced by: sxbrsigalem4 31777

W3C validator