sxbrsigalem2 - Mathbox for Thierry Arnoux

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Theorem sxbrsigalem2 34254

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 34252	. . 3 ⊢ ∪ ran 𝑅 = (ℝ × ℝ)
5		sxbrsigalem0 34239	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
6	4, 5	eqtr4i 2755	. 2 ⊢ ∪ ran 𝑅 = ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
7		vex 3440	. . . . . 6 ⊢ 𝑢 ∈ V
8		vex 3440	. . . . . 6 ⊢ 𝑣 ∈ V
9	7, 8	xpex 7689	. . . . 5 ⊢ (𝑢 × 𝑣) ∈ V
10	3, 9	elrnmpo 7485	. . . 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 34244	. . . . . . . . . . . . 13 ⊢ ran 𝐼 ⊆ 𝔅_ℝ
13		brsigasspwrn 34152	. . . . . . . . . . . . 13 ⊢ 𝔅_ℝ ⊆ 𝒫 ℝ
14	12, 13	sstri 3945	. . . . . . . . . . . 12 ⊢ ran 𝐼 ⊆ 𝒫 ℝ
15	14	sseli 3931	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ∈ 𝒫 ℝ)
16	15	elpwid 4560	. . . . . . . . . 10 ⊢ (𝑢 ∈ ran 𝐼 → 𝑢 ⊆ ℝ)
17	14	sseli 3931	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ∈ 𝒫 ℝ)
18	17	elpwid 4560	. . . . . . . . . 10 ⊢ (𝑣 ∈ ran 𝐼 → 𝑣 ⊆ ℝ)
19		xpinpreima2 33874	. . . . . . . . . 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 11100	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
22	21	mptex 7159	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
23	22	rnex 7843	. . . . . . . . . . . . . . 15 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∈ V
24	21	mptex 7159	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
25	24	rnex 7843	. . . . . . . . . . . . . . 15 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ∈ V
26	23, 25	unex 7680	. . . . . . . . . . . . . 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 34109	. . . . . . . . . . . 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 32649	. . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) ∈ V
34	2, 33	elrnmpo 7485	. . . . . . . . . . . . 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 5648	. . . . . . . . . . . . . . . 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 12580	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ∈ ℝ)
39		2rp 12898	. . . . . . . . . . . . . . . . . . . . . . . . 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 14154	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (2↑𝑛) ∈ ℝ⁺)
43	38, 42	rerpdivcld 12968	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ)
44	43	rexrd 11165	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ∈ ℝ^*)
45		1red 11116	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 1 ∈ ℝ)
46	38, 45	readdcld 11144	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 + 1) ∈ ℝ)
47	46, 42	rerpdivcld 12968	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ)
48	47	rexrd 11165	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑥 + 1) / (2↑𝑛)) ∈ ℝ^*)
49		pnfxr 11169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ +∞ ∈ ℝ^*
50	49	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → +∞ ∈ ℝ^*)
51	38	lep1d 12056	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑥 ≤ (𝑥 + 1))
52	38, 46, 42, 51	lediv1dd 12995	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑥 / (2↑𝑛)) ≤ ((𝑥 + 1) / (2↑𝑛)))
53		pnfge 13032	. . . . . . . . . . . . . . . . . . . . . 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 32726	. . . . . . . . . . . . . . . . . . . . 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 5648	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = (((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))) × ℝ))
58		difxp1 6114	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞)) × ℝ) = ((((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∖ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
59	57, 58	eqtr3di 2779	. . . . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
62		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ)
63		oveq1 7356	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → (𝑒[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
64	63	xpeq1d 5648	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = (𝑥 / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = (((𝑥 / (2↑𝑛))[,)+∞) × ℝ))
65	64	rspceeqv 3600	. . . . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
68		ovex 7382	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒[,)+∞) ∈ V
69	68, 21	xpex 7689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
70	67, 69	elrnmpti 5904	. . . . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝑥 / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
73		elsigagen 34114	. . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ)
76		oveq1 7356	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → (𝑒[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
77	76	xpeq1d 5648	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = ((𝑥 + 1) / (2↑𝑛)) → ((𝑒[,)+∞) × ℝ) = ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ))
78	77	rspceeqv 3600	. . . . . . . . . . . . . . . . . . . . . . 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 5904	. . . . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((((𝑥 + 1) / (2↑𝑛))[,)+∞) × ℝ) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
83		elsigagen 34114	. . . . . . . . . . . . . . . . . . . 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 34100	. . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . 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 2828	. . . . . . . . . . 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 32650	. . . . . . . . . . . . 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 7485	. . . . . . . . . . . . 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 5649	. . . . . . . . . . . . . . . 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 5649	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))))
101		difxp2 6115	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × (((𝑥 / (2↑𝑛))[,)+∞) ∖ (((𝑥 + 1) / (2↑𝑛))[,)+∞))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
102	100, 101	eqtr3di 2779	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) = ((ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∖ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))))
103		ssun2 4130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
104		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞))
105		oveq1 7356	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (𝑓[,)+∞) = ((𝑥 / (2↑𝑛))[,)+∞))
106	105	xpeq2d 5649	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = (𝑥 / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)))
107	106	rspceeqv 3600	. . . . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
110		ovex 7382	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓[,)+∞) ∈ V
111	21, 110	xpex 7689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
112	109, 111	elrnmpti 5904	. . . . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × ((𝑥 / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
115		elsigagen 34114	. . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞))
118		oveq1 7356	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (𝑓[,)+∞) = (((𝑥 + 1) / (2↑𝑛))[,)+∞))
119	118	xpeq2d 5649	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 = ((𝑥 + 1) / (2↑𝑛)) → (ℝ × (𝑓[,)+∞)) = (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)))
120	119	rspceeqv 3600	. . . . . . . . . . . . . . . . . . . . . . 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 5904	. . . . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (ℝ × (((𝑥 + 1) / (2↑𝑛))[,)+∞)) ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
125		elsigagen 34114	. . . . . . . . . . . . . . . . . . . 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 34100	. . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . 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 2828	. . . . . . . . . . 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 34102	. . . . . . . . . 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 2828	. . . . . . . 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 2828	. . . . . 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 3171	. . . 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 3939	. 2 ⊢ ran 𝑅 ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
146		sigagenss2 34117	. 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 2109 ∃wrex 3053 Vcvv 3436 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 𝒫 cpw 4551 ∪ cuni 4858 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ^◡ccnv 5618 ran crn 5620 ↾ cres 5621 “ cima 5622 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 1^st c1st 7922 2^nd c2nd 7923 ℝcr 11008 1c1 11010 + caddc 11012 +∞cpnf 11146 ℝ^*cxr 11148 ≤ cle 11150 / cdiv 11777 2c2 12183 ℤcz 12471 ℝ⁺crp 12893 (,)cioo 13248 [,)cico 13250 ↑cexp 13968 topGenctg 17341 sigAlgebracsiga 34075 sigaGencsigagen 34105 𝔅_ℝcbrsiga 34148

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-ac2 10357 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-ac 10010 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-refld 21512 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-fcls 23826 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-cfil 25153 df-cmet 25155 df-cms 25233 df-limc 25765 df-dv 25766 df-log 26463 df-cxp 26464 df-logb 26673 df-siga 34076 df-sigagen 34106 df-brsiga 34149

This theorem is referenced by: sxbrsigalem4 34255

W3C validator