sxbrsigalem2 - Mathbox for Thierry Arnoux

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Theorem sxbrsigalem2 30348

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 30346	. . 3 ⊢ ∪ ran 𝑅 = (ℝ × ℝ)
5		sxbrsigalem0 30333	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
6	4, 5	eqtr4i 2647	. 2 ⊢ ∪ ran 𝑅 = ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
7		vex 3203	. . . . . 6 ⊢ 𝑢 ∈ V
8		vex 3203	. . . . . 6 ⊢ 𝑣 ∈ V
9	7, 8	xpex 6962	. . . . 5 ⊢ (𝑢 × 𝑣) ∈ V
10	3, 9	elrnmpt2 6773	. . . 4 ⊢ (𝑑 ∈ ran 𝑅 ↔ ∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼 𝑑 = (𝑢 × 𝑣))
11		simpr 477	. . . . . . 7 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑑 = (𝑢 × 𝑣)) → 𝑑 = (𝑢 × 𝑣))
12	1, 2	dya2icobrsiga 30338	. . . . . . . . . . . . 13 ⊢ ran 𝐼 ⊆ 𝔅_ℝ
13		brsigasspwrn 30248	. . . . . . . . . . . . 13 ⊢ 𝔅_ℝ ⊆ 𝒫 ℝ
14	12, 13	sstri 3612	. . . . . . . . . . . 12 ⊢ ran 𝐼 ⊆ 𝒫 ℝ
15	14	sseli 3599	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ∈ 𝒫 ℝ)
16	15	elpwid 4170	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ⊆ ℝ)
17	14	sseli 3599	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ∈ 𝒫 ℝ)
18	17	elpwid 4170	. . . . . . . . . 10 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ⊆ ℝ)
19		xpinpreima2 29953	. . . . . . . . . 10 ⊢ ((𝑢 ⊆ ℝ ∧ 𝑣 ⊆ ℝ) → (𝑢 × 𝑣) = ((^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∩ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣)))
20	16, 18, 19	syl2an 494	. . . . . . . . 9 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 × 𝑣) = ((^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∩ (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣)))
21		reex 10027	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
22	21	mptex 6486	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
23	22	rnex 7100	. . . . . . . . . . . . . . 15 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
24	21	mptex 6486	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
25	24	rnex 7100	. . . . . . . . . . . . . . 15 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
26	23, 25	unex 6956	. . . . . . . . . . . . . 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 30205	. . . . . . . . . . . 12 ⊢ (⊤ → (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ∈ ∪ ran sigAlgebra)
29	28	trud 1493	. . . . . . . . . . 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 29484	. . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V
34	2, 33	elrnmpt2 6773	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ ran 𝐼 ↔ ∃𝑥 ∈ ℤ ∃𝑛 ∈ ℤ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
35		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → 𝑢 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
36	35	xpeq1d 5138	. . . . . . . . . . . . . . . 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 5559	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∖ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
38		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℤ)
39	38	zred 11482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
40		2rp 11837	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 2 ∈ ℝ⁺
41	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 2 ∈ ℝ⁺)
42		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
43	41, 42	rpexpcld 13032	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2↑𝑛) ∈ ℝ⁺)
44	39, 43	rerpdivcld 11903	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ)
45	44	rexrd 10089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ^*)
46		1red 10055	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 1 ∈ ℝ)
47	39, 46	readdcld 10069	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 + 1) ∈ ℝ)
48	47, 43	rerpdivcld 11903	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ)
49	48	rexrd 10089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ^*)
50		pnfxr 10092	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ +∞ ∈ ℝ^*
51	50	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → +∞ ∈ ℝ^*)
52	39	lep1d 10955	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ≤ (𝑥 + 1))
53	39, 47, 43, 52	lediv1dd 11930	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ≤ ((𝑥 + 1) / (2↑𝑛)))
54		pnfge 11964	. . . . . . . . . . . . . . . . . . . . . 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 29545	. . . . . . . . . . . . . . . . . . . . 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 1334	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
58	57	xpeq1d 5138	. . . . . . . . . . . . . . . . . . 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 2671	. . . . . . . . . . . . . . . . . 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 3776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
62		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)
63		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → (𝑒[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
64	63	xpeq1d 5138	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ))
65	64	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ) ↔ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)))
66	65	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛)) ∈ ℝ ∧ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)) → ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
67	44, 62, 66	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
68		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
69		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒[,)+∞) ∈ V
70	69, 21	xpex 6962	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
71	68, 70	elrnmpti 5376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ↔ ∃𝑒 ∈ ℝ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
72	67, 71	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
73	61, 72	sseldi 3601	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
74		elsigagen 30210	. . . . . . . . . . . . . . . . . . . 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[,)+∞))))))
75	26, 73, 74	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
76		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)
77		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → (𝑒[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
78	77	xpeq1d 5138	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
79	78	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → (((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ) ↔ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)))
80	79	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 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))
81	48, 76, 80	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑒 ∈ ℝ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
82	68, 70	elrnmpti 5376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ↔ ∃𝑒 ∈ ℝ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((𝑒[,)+∞) × ℝ))
83	81, 82	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
84	61, 83	sseldi 3601	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
85		elsigagen 30210	. . . . . . . . . . . . . . . . . . . 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[,)+∞))))))
86	26, 84, 85	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
87		difelsiga 30196	. . . . . . . . . . . . . . . . . . 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[,)+∞))))))
88	60, 75, 86, 87	syl3anc 1326	. . . . . . . . . . . . . . . . . 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[,)+∞))))))
89	59, 88	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
90	89	adantr 481	. . . . . . . . . . . . . . . 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[,)+∞))))))
91	36, 90	eqeltrd 2701	. . . . . . . . . . . . . . 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[,)+∞))))))
92	91	ex 450	. . . . . . . . . . . . . 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[,)+∞)))))))
93	92	rexlimivv 3036	. . . . . . . . . . . . 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[,)+∞))))))
94	34, 93	sylbi 207	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ran 𝐼 → (𝑢 × ℝ) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
95	32, 94	eqeltrd 2701	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran 𝐼 → (^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
96	95	adantr 481	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (^◡(1^st ↾ (ℝ × ℝ)) “ 𝑢) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
97		2ndpreima 29485	. . . . . . . . . . . . 13 ⊢ (𝑣 ⊆ ℝ → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) = (ℝ × 𝑣))
98	18, 97	syl 17	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran 𝐼 → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) = (ℝ × 𝑣))
99	2, 33	elrnmpt2 6773	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran 𝐼 ↔ ∃𝑥 ∈ ℤ ∃𝑛 ∈ ℤ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
100		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))
101	100	xpeq2d 5139	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑣 = ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) → (ℝ × 𝑣) = (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))))
102		difxp2 5560	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
103	57	xpeq2d 5139	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))))
104	102, 103	syl5reqr 2671	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))))
105		ssun2 3777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
106		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))
107		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (𝑓[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
108	107	xpeq2d 5139	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)))
109	108	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)) ↔ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))))
110	109	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛)) ∈ ℝ ∧ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))) → ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
111	44, 106, 110	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
112		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
113		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓[,)+∞) ∈ V
114	21, 113	xpex 6962	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
115	112, 114	elrnmpti 5376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ↔ ∃𝑓 ∈ ℝ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
116	111, 115	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
117	105, 116	sseldi 3601	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
118		elsigagen 30210	. . . . . . . . . . . . . . . . . . . 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[,)+∞))))))
119	26, 117, 118	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
120		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))
121		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (𝑓[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
122	121	xpeq2d 5139	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
123	122	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → ((ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)) ↔ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))))
124	123	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 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[,)+∞)))
125	48, 120, 124	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ∃𝑓 ∈ ℝ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
126	112, 114	elrnmpti 5376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ↔ ∃𝑓 ∈ ℝ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (𝑓[,)+∞)))
127	125, 126	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
128	105, 127	sseldi 3601	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
129		elsigagen 30210	. . . . . . . . . . . . . . . . . . . 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[,)+∞))))))
130	26, 128, 129	sylancr 695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
131		difelsiga 30196	. . . . . . . . . . . . . . . . . . 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[,)+∞))))))
132	60, 119, 130, 131	syl3anc 1326	. . . . . . . . . . . . . . . . . 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[,)+∞))))))
133	104, 132	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
134	133	adantr 481	. . . . . . . . . . . . . . . 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[,)+∞))))))
135	101, 134	eqeltrd 2701	. . . . . . . . . . . . . . 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[,)+∞))))))
136	135	ex 450	. . . . . . . . . . . . . 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[,)+∞)))))))
137	136	rexlimivv 3036	. . . . . . . . . . . . 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[,)+∞))))))
138	99, 137	sylbi 207	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran 𝐼 → (ℝ × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
139	98, 138	eqeltrd 2701	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐼 → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
140	139	adantl 482	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (^◡(2^nd ↾ (ℝ × ℝ)) “ 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
141		inelsiga 30198	. . . . . . . . . 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[,)+∞))))))
142	30, 96, 140, 141	syl3anc 1326	. . . . . . . . 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[,)+∞))))))
143	20, 142	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑢 × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
144	143	adantr 481	. . . . . . 7 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑑 = (𝑢 × 𝑣)) → (𝑢 × 𝑣) ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
145	11, 144	eqeltrd 2701	. . . . . 6 ⊢ (((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) ∧ 𝑑 = (𝑢 × 𝑣)) → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
146	145	ex 450	. . . . 5 ⊢ ((𝑢 ∈ ran 𝐼 ∧ 𝑣 ∈ ran 𝐼) → (𝑑 = (𝑢 × 𝑣) → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))))
147	146	rexlimivv 3036	. . . 4 ⊢ (∃𝑢 ∈ ran 𝐼∃𝑣 ∈ ran 𝐼 𝑑 = (𝑢 × 𝑣) → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
148	10, 147	sylbi 207	. . 3 ⊢ (𝑑 ∈ ran 𝑅 → 𝑑 ∈ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))))
149	148	ssriv 3607	. 2 ⊢ ran 𝑅 ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
150		sigagenss2 30213	. 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[,)+∞))))))
151	6, 149, 26, 150	mp3an 1424	1 ⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ^◡ccnv 5113 ran crn 5115 ↾ cres 5116 “ cima 5117 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1^st c1st 7166 2^nd c2nd 7167 ℝcr 9935 1c1 9937 + caddc 9939 +∞cpnf 10071 ℝ^*cxr 10073 ≤ cle 10075 / cdiv 10684 2c2 11070 ℤcz 11377 ℝ⁺crp 11832 (,)cioo 12175 [,)cico 12177 ↑cexp 12860 topGenctg 16098 sigAlgebracsiga 30170 sigaGencsigagen 30201 𝔅_ℝcbrsiga 30244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-refld 19951 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-fcls 21745 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-cfil 23053 df-cmet 23055 df-cms 23132 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-logb 24503 df-siga 30171 df-sigagen 30202 df-brsiga 30245

This theorem is referenced by: sxbrsigalem4 30349

W3C validator