sxbrsigalem2 - Mathbox for Thierry Arnoux

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Theorem sxbrsigalem2 34264

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 34262	. . 3 ⊢ ∪ ran 𝑅 = (ℝ × ℝ)
5		sxbrsigalem0 34249	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
6	4, 5	eqtr4i 2761	. 2 ⊢ ∪ ran 𝑅 = ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
7		vex 3463	. . . . . 6 ⊢ 𝑢 ∈ V
8		vex 3463	. . . . . 6 ⊢ 𝑣 ∈ V
9	7, 8	xpex 7745	. . . . 5 ⊢ (𝑢 × 𝑣) ∈ V
10	3, 9	elrnmpo 7541	. . . 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 34254	. . . . . . . . . . . . 13 ⊢ ran 𝐼 ⊆ 𝔅_ℝ
13		brsigasspwrn 34162	. . . . . . . . . . . . 13 ⊢ 𝔅_ℝ ⊆ 𝒫 ℝ
14	12, 13	sstri 3968	. . . . . . . . . . . 12 ⊢ ran 𝐼 ⊆ 𝒫 ℝ
15	14	sseli 3954	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ∈ 𝒫 ℝ)
16	15	elpwid 4584	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ⊆ ℝ)
17	14	sseli 3954	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ∈ 𝒫 ℝ)
18	17	elpwid 4584	. . . . . . . . . 10 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ⊆ ℝ)
19		xpinpreima2 33884	. . . . . . . . . 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 11218	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
22	21	mptex 7214	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
23	22	rnex 7904	. . . . . . . . . . . . . . 15 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
24	21	mptex 7214	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
25	24	rnex 7904	. . . . . . . . . . . . . . 15 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
26	23, 25	unex 7736	. . . . . . . . . . . . . 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 34119	. . . . . . . . . . . 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 32630	. . . . . . . . . . . . 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 7436	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V
34	2, 33	elrnmpo 7541	. . . . . . . . . . . . 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 5683	. . . . . . . . . . . . . . . 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 12695	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
39		2rp 13011	. . . . . . . . . . . . . . . . . . . . . . . . 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 14263	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2↑𝑛) ∈ ℝ⁺)
43	38, 42	rerpdivcld 13080	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ)
44	43	rexrd 11283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ^*)
45		1red 11234	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 1 ∈ ℝ)
46	38, 45	readdcld 11262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 + 1) ∈ ℝ)
47	46, 42	rerpdivcld 13080	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ)
48	47	rexrd 11283	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ^*)
49		pnfxr 11287	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ +∞ ∈ ℝ^*
50	49	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → +∞ ∈ ℝ^*)
51	38	lep1d 12171	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ≤ (𝑥 + 1))
52	38, 46, 42, 51	lediv1dd 13107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ≤ ((𝑥 + 1) / (2↑𝑛)))
53		pnfge 13144	. . . . . . . . . . . . . . . . . . . . . 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 32706	. . . . . . . . . . . . . . . . . . . . 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 5683	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ))
58		difxp1 6154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∖ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
59	57, 58	eqtr3di 2785	. . . . . . . . . . . . . . . . . 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 4153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
62		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)
63		oveq1 7410	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → (𝑒[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
64	63	xpeq1d 5683	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ))
65	64	rspceeqv 3624	. . . . . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
68		ovex 7436	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒[,)+∞) ∈ V
69	68, 21	xpex 7745	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
70	67, 69	elrnmpti 5942	. . . . . . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
73		elsigagen 34124	. . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)
76		oveq1 7410	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → (𝑒[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
77	76	xpeq1d 5683	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
78	77	rspceeqv 3624	. . . . . . . . . . . . . . . . . . . . . . 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 5942	. . . . . . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
83		elsigagen 34124	. . . . . . . . . . . . . . . . . . . 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 34110	. . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . 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 2834	. . . . . . . . . . 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 32631	. . . . . . . . . . . . 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 7541	. . . . . . . . . . . . 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 5684	. . . . . . . . . . . . . . . 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 5684	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))))
101		difxp2 6155	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
102	100, 101	eqtr3di 2785	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))))
103		ssun2 4154	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
104		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))
105		oveq1 7410	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (𝑓[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
106	105	xpeq2d 5684	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)))
107	106	rspceeqv 3624	. . . . . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
110		ovex 7436	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓[,)+∞) ∈ V
111	21, 110	xpex 7745	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
112	109, 111	elrnmpti 5942	. . . . . . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
115		elsigagen 34124	. . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))
118		oveq1 7410	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (𝑓[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
119	118	xpeq2d 5684	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
120	119	rspceeqv 3624	. . . . . . . . . . . . . . . . . . . . . . 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 5942	. . . . . . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
125		elsigagen 34124	. . . . . . . . . . . . . . . . . . . 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 34110	. . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . 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 2834	. . . . . . . . . . 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 34112	. . . . . . . . . 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 2834	. . . . . . . 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 2834	. . . . . 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 3186	. . . 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 3962	. 2 ⊢ ran 𝑅 ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
146		sigagenss2 34127	. 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 3060 Vcvv 3459 ∖ cdif 3923 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 𝒫 cpw 4575 ∪ cuni 4883 class class class wbr 5119 ↦ cmpt 5201 × cxp 5652 ^◡ccnv 5653 ran crn 5655 ↾ cres 5656 “ cima 5657 ‘cfv 6530 (class class class)co 7403 ∈ cmpo 7405 1^st c1st 7984 2^nd c2nd 7985 ℝcr 11126 1c1 11128 + caddc 11130 +∞cpnf 11264 ℝ^*cxr 11266 ≤ cle 11268 / cdiv 11892 2c2 12293 ℤcz 12586 ℝ⁺crp 13006 (,)cioo 13360 [,)cico 13362 ↑cexp 14077 topGenctg 17449 sigAlgebracsiga 34085 sigaGencsigagen 34115 𝔅_ℝcbrsiga 34158

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-ac2 10475 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-dju 9913 df-card 9951 df-acn 9954 df-ac 10128 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-sum 15701 df-ef 16081 df-sin 16083 df-cos 16084 df-pi 16086 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-refld 21563 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-cmp 23323 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-fcls 23877 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-cfil 25205 df-cmet 25207 df-cms 25285 df-limc 25817 df-dv 25818 df-log 26515 df-cxp 26516 df-logb 26725 df-siga 34086 df-sigagen 34116 df-brsiga 34159

This theorem is referenced by: sxbrsigalem4 34265

W3C validator