sxbrsigalem2 - Mathbox for Thierry Arnoux

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Theorem sxbrsigalem2 34288

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 34286	. . 3 ⊢ ∪ ran 𝑅 = (ℝ × ℝ)
5		sxbrsigalem0 34273	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
6	4, 5	eqtr4i 2768	. 2 ⊢ ∪ ran 𝑅 = ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
7		vex 3484	. . . . . 6 ⊢ 𝑢 ∈ V
8		vex 3484	. . . . . 6 ⊢ 𝑣 ∈ V
9	7, 8	xpex 7773	. . . . 5 ⊢ (𝑢 × 𝑣) ∈ V
10	3, 9	elrnmpo 7569	. . . 4 ⊢ (𝑑 ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼 𝑑 = (𝑢 × 𝑣))
11		simpr 484	. . . . . . 7 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑑 = (𝑢 × 𝑣)) → 𝑑 = (𝑢 × 𝑣))
12	1, 2	dya2icobrsiga 34278	. . . . . . . . . . . . 13 ⊢ ran 𝐼 ⊆ 𝔅_ℝ
13		brsigasspwrn 34186	. . . . . . . . . . . . 13 ⊢ 𝔅_ℝ ⊆ 𝒫 ℝ
14	12, 13	sstri 3993	. . . . . . . . . . . 12 ⊢ ran 𝐼 ⊆ 𝒫 ℝ
15	14	sseli 3979	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ∈ 𝒫 ℝ)
16	15	elpwid 4609	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ⊆ ℝ)
17	14	sseli 3979	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ∈ 𝒫 ℝ)
18	17	elpwid 4609	. . . . . . . . . 10 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ⊆ ℝ)
19		xpinpreima2 33906	. . . . . . . . . 10 ⊢ ((𝑢 ⊆ ℝ ∧ 𝑣 ⊆ ℝ) → (𝑢 × 𝑣) = ((^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∩ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣)))
20	16, 18, 19	syl2an 596	. . . . . . . . 9 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 × 𝑣) = ((^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∩ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣)))
21		reex 11246	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
22	21	mptex 7243	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
23	22	rnex 7932	. . . . . . . . . . . . . . 15 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
24	21	mptex 7243	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
25	24	rnex 7932	. . . . . . . . . . . . . . 15 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
26	23, 25	unex 7764	. . . . . . . . . . . . . 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 34143	. . . . . . . . . . . 12 ⊢ (⊤ → (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra)
29	28	mptru 1547	. . . . . . . . . . 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 32716	. . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V
34	2, 33	elrnmpo 7569	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ran 𝐼 ↔ ∃𝑥 ∈ ℤ ∃𝑛 ∈ ℤ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
35		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
36	35	xpeq1d 5714	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (𝑢 × ℝ) = (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ))
37		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℤ)
38	37	zred 12722	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
39		2rp 13039	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ⁺
40	39	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 2 ∈ ℝ⁺)
41		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
42	40, 41	rpexpcld 14286	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2↑𝑛) ∈ ℝ⁺)
43	38, 42	rerpdivcld 13108	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ)
44	43	rexrd 11311	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ^*)
45		1red 11262	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 1 ∈ ℝ)
46	38, 45	readdcld 11290	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 + 1) ∈ ℝ)
47	46, 42	rerpdivcld 13108	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ)
48	47	rexrd 11311	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ^*)
49		pnfxr 11315	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ +∞ ∈ ℝ^*
50	49	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → +∞ ∈ ℝ^*)
51	38	lep1d 12199	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ≤ (𝑥 + 1))
52	38, 46, 42, 51	lediv1dd 13135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ≤ ((𝑥 + 1) / (2↑𝑛)))
53		pnfge 13172	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 + 1) / (2↑𝑛)) ∈ ℝ^* → ((𝑥 + 1) / (2↑𝑛)) ≤ +∞)
54	48, 53	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ≤ +∞)
55		difico 32785	. . . . . . . . . . . . . . . . . . . . 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))))
56	44, 48, 50, 52, 54, 55	syl32anc 1380	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
57	56	xpeq1d 5714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ))
58		difxp1 6185	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∖ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
59	57, 58	eqtr3di 2792	. . . . . . . . . . . . . . . . . 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 4178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
62		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)
63		oveq1 7438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → (𝑒[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
64	63	xpeq1d 5714	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ))
65	64	rspceeqv 3645	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛)) ∈ ℝ ∧ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)) → ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
66	43, 62, 65	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
67		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
68		ovex 7464	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒[,)+∞) ∈ V
69	68, 21	xpex 7773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
70	67, 69	elrnmpti 5973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ↔ ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
71	66, 70	sylibr 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
72	61, 71	sselid 3981	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
73		elsigagen 34148	. . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
75		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)
76		oveq1 7438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → (𝑒[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
77	76	xpeq1d 5714	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
78	77	rspceeqv 3645	. . . . . . . . . . . . . . . . . . . . . . 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	47, 75, 78	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑒 ∈ ℝ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
80	67, 69	elrnmpti 5973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ↔ ∃𝑒 ∈ ℝ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
81	79, 80	sylibr 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
82	61, 81	sselid 3981	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
83		elsigagen 34148	. . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
85		difelsiga 34134	. . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . 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 217	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ran 𝐼 → (𝑢 × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
93	32, 92	eqeltrd 2841	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran 𝐼 → (^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
94	93	adantr 480	. . . . . . . . . 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 32717	. . . . . . . . . . . . 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 7569	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran 𝐼 ↔ ∃𝑥 ∈ ℤ ∃𝑛 ∈ ℤ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
98		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
99	98	xpeq2d 5715	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (ℝ × 𝑣) = (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))))
100	56	xpeq2d 5715	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))))
101		difxp2 6186	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
102	100, 101	eqtr3di 2792	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))))
103		ssun2 4179	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
104		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))
105		oveq1 7438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (𝑓[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
106	105	xpeq2d 5715	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)))
107	106	rspceeqv 3645	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛)) ∈ ℝ ∧ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))) → ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
108	43, 104, 107	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
109		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
110		ovex 7464	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓[,)+∞) ∈ V
111	21, 110	xpex 7773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
112	109, 111	elrnmpti 5973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ↔ ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
113	108, 112	sylibr 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
114	103, 113	sselid 3981	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
115		elsigagen 34148	. . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
117		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))
118		oveq1 7438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (𝑓[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
119	118	xpeq2d 5715	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
120	119	rspceeqv 3645	. . . . . . . . . . . . . . . . . . . . . . 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	47, 117, 120	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑓 ∈ ℝ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
122	109, 111	elrnmpti 5973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ↔ ∃𝑓 ∈ ℝ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
123	121, 122	sylibr 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
124	103, 123	sselid 3981	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
125		elsigagen 34148	. . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
127		difelsiga 34134	. . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 3201	. . . . . . . . . . . . 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 217	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran 𝐼 → (ℝ × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
135	96, 134	eqeltrd 2841	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐼 → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
136	135	adantl 481	. . . . . . . . . 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 34136	. . . . . . . . . 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 1373	. . . . . . . . 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 2841	. . . . . . . 8 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
140	139	adantr 480	. . . . . . 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 2841	. . . . . 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 412	. . . . 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 3201	. . . 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 217	. . 3 ⊢ (𝑑 ∈ ran 𝑅 → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
145	144	ssriv 3987	. 2 ⊢ ran 𝑅 ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
146		sigagenss2 34151	. 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 1463	1 ⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ^◡ccnv 5684 ran crn 5686 ↾ cres 5687 “ cima 5688 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1^st c1st 8012 2^nd c2nd 8013 ℝcr 11154 1c1 11156 + caddc 11158 +∞cpnf 11292 ℝ^*cxr 11294 ≤ cle 11296 / cdiv 11920 2c2 12321 ℤcz 12613 ℝ⁺crp 13034 (,)cioo 13387 [,)cico 13389 ↑cexp 14102 topGenctg 17482 sigAlgebracsiga 34109 sigaGencsigagen 34139 𝔅_ℝcbrsiga 34182

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-refld 21623 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-fcls 23949 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-cfil 25289 df-cmet 25291 df-cms 25369 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 df-logb 26808 df-siga 34110 df-sigagen 34140 df-brsiga 34183

This theorem is referenced by: sxbrsigalem4 34289

W3C validator