meaiuninc3v - Mathbox for Glauco Siliprandi

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Theorem meaiuninc3v 43993

Description: Measures are continuous from below: if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is the general case of Proposition 112C (e) of [Fremlin1] p. 16 . This theorem generalizes meaiuninc 43990 and meaiuninc2 43991 where the sequence is required to be bounded. (Contributed by Glauco Siliprandi, 13-Feb-2022.)

**Hypotheses**
Ref	Expression
meaiuninc3v.m	⊢ (𝜑 → 𝑀 ∈ Meas)
meaiuninc3v.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
meaiuninc3v.z	⊢ 𝑍 = (ℤ_≥‘𝑁)
meaiuninc3v.e	⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
meaiuninc3v.i	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
meaiuninc3v.s	⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))

**Assertion**
Ref	Expression
meaiuninc3v	⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Distinct variable groups: 𝑛,𝐸 𝑛,𝑀 𝑛,𝑍 𝜑,𝑛 Allowed substitution hints: 𝑆(𝑛) 𝑁(𝑛)

Proof of Theorem meaiuninc3v Dummy variables 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		meaiuninc3v.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
2	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑁 ∈ ℤ)
3		meaiuninc3v.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑁)
4		meaiuninc3v.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Meas)
5	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
6		eqid 2740	. . . . . 6 ⊢ dom 𝑀 = dom 𝑀
7		meaiuninc3v.e	. . . . . . 7 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
8	7	ffvelrnda 6958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
9	5, 6, 8	meaxrcl 43970	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
10		meaiuninc3v.s	. . . . 5 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
11	9, 10	fmptd 6985	. . . 4 ⊢ (𝜑 → 𝑆:𝑍⟶ℝ^*)
12	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆:𝑍⟶ℝ^*)
13		nfv 1921	. . . . 5 ⊢ Ⅎ𝑛𝜑
14		nfcv 2909	. . . . . 6 ⊢ Ⅎ𝑛ℝ
15		nfra1 3145	. . . . . 6 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥
16	14, 15	nfrex 3240	. . . . 5 ⊢ Ⅎ𝑛∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥
17	13, 16	nfan 1906	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
18		nfcv 2909	. . . 4 ⊢ Ⅎ𝑛𝐸
19	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑀 ∈ Meas)
20	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝐸:𝑍⟶dom 𝑀)
21		meaiuninc3v.i	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
22	21	adantlr 712	. . . 4 ⊢ (((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
23		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
24	17, 18, 19, 2, 3, 20, 22, 23, 10	meaiunincf 43992	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
25	2, 3, 12, 24	climxlim2 43358	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
26		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
27		2fveq3 6776	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑛 → (𝑀‘(𝐸‘𝑗)) = (𝑀‘(𝐸‘𝑛)))
28	27	breq2d 5091	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ 𝑥 < (𝑀‘(𝐸‘𝑛))))
29	28	cbvrexvw 3382	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∃𝑛 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑛)))
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∃𝑛 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑛))))
31		rexr 11022	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
32	31	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ℝ^*)
33	9	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
34	32, 33	xrltnled 42873	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑥 < (𝑀‘(𝐸‘𝑛)) ↔ ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
35	34	rexbidva 3227	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑛)) ↔ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
36	30, 35	bitrd 278	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
37	36	ralbidva 3122	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
38		rexnal 3168	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
39	38	ralbii 3093	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
40		ralnex 3166	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
41	39, 40	bitri 274	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
42	41	a1i 11	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
43	37, 42	bitrd 278	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
44	43	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
45	26, 44	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
46		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
47	45, 46	syldan 591	. . 3 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
48		simp-4r 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ)
49	48, 31	syl 17	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ^*)
50		simp-4l 780	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝜑)
51	3	uztrn2 12600	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
52	51	ad4ant24 751	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
53	11	ffvelrnda 6958	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) ∈ ℝ^*)
54	50, 52, 53	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) ∈ ℝ^*)
55		eleq1w 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑛 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
56	55	anbi2d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
57		2fveq3 6776	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑗)))
58	57	eleq1d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝑀‘(𝐸‘𝑛)) ∈ ℝ^* ↔ (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*))
59	56, 58	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^)))
60	59, 9	chvarvv 2006	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
61	60	ad5ant13 754	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
62		simplr 766	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 < (𝑀‘(𝐸‘𝑗)))
63	4	3ad2ant1 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑀 ∈ Meas)
64	7	ffvelrnda 6958	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ dom 𝑀)
65	64	3adant3 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑗) ∈ dom 𝑀)
66		simp1 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝜑)
67	51	3adant1 1129	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
68	66, 67, 8	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑛) ∈ dom 𝑀)
69		simp3 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ (ℤ_≥‘𝑗))
70		simpll 764	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝜑)
71	3	uzssd3 42937	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
72	71	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → (ℤ_≥‘𝑗) ⊆ 𝑍)
73		elfzouz 13390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (𝑗..^𝑛) → 𝑘 ∈ (ℤ_≥‘𝑗))
74	73	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ (ℤ_≥‘𝑗))
75	72, 74	sseldd 3927	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ 𝑍)
76	75	adantll 711	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ 𝑍)
77		eleq1w 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍))
78	77	anbi2d 629	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍)))
79		fveq2 6771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘))
80		fvoveq1 7294	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1)))
81	79, 80	sseq12d 3959	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))))
82	78, 81	imbi12d 345	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))))
83	82, 21	chvarvv 2006	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
84	70, 76, 83	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
85	84	3adantl3 1167	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) ∧ 𝑘 ∈ (𝑗..^𝑛)) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
86	69, 85	ssinc 42607	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑗) ⊆ (𝐸‘𝑛))
87	63, 6, 65, 68, 86	meassle 43972	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘(𝐸‘𝑛)))
88		fvexd 6786	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑛)) ∈ V)
89	10	fvmpt2 6883	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝑀‘(𝐸‘𝑛)) ∈ V) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
90	51, 88, 89	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
91	90	3adant1 1129	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
92	87, 91	breqtrrd 5107	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑆‘𝑛))
93	92	ad5ant135 1367	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑆‘𝑛))
94	49, 61, 54, 62, 93	xrltletrd 12894	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 < (𝑆‘𝑛))
95	49, 54, 94	xrltled 12883	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 ≤ (𝑆‘𝑛))
96	95	ralrimiva 3110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛))
97	96	ex 413	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝑥 < (𝑀‘(𝐸‘𝑗)) → ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
98	97	reximdva 3205	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
99	98	ralimdva 3105	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
100	99	imp 407	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛))
101		nfmpt1 5187	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
102	10, 101	nfcxfr 2907	. . . . . . 7 ⊢ Ⅎ𝑛𝑆
103	102, 1, 3, 11	xlimpnf 43354	. . . . . 6 ⊢ (𝜑 → (𝑆~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
104	103	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑆~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
105	100, 104	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑆~~>*+∞)
106		nfv 1921	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
107		nfra1 3145	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
108	106, 107	nfan 1906	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
109		rspa 3133	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
110	109	adantll 711	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
111		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝜑
112		nfcv 2909	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ℝ
113		nfre1 3237	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
114	112, 113	nfralw 3152	. . . . . . . . . 10 ⊢ Ⅎ𝑗∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
115	111, 114	nfan 1906	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
116		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
117	115, 116	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ)
118		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
119	31	ad3antlr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 ∈ ℝ^*)
120	4, 6	dmmeasal 43961	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
121	3	uzct 42581	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ≼ ω
122	121	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ≼ ω)
123	120, 122, 8	saliuncl 43834	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ dom 𝑀)
124	4, 6, 123	meaxrcl 43970	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
125	124	ad3antrrr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
126	60	ad4ant13 748	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
127		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 < (𝑀‘(𝐸‘𝑗)))
128	4	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑀 ∈ Meas)
129	123	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ dom 𝑀)
130		fveq2 6771	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝐸‘𝑛) = (𝐸‘𝑗))
131	130	ssiun2s 4983	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (𝐸‘𝑗) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
132	131	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
133	128, 6, 64, 129, 132	meassle 43972	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
134	133	ad4ant13 748	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
135	119, 126, 125, 127, 134	xrltletrd 12894	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 < (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
136	119, 125, 135	xrltled 12883	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
137	136	exp31 420	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑗 ∈ 𝑍 → (𝑥 < (𝑀‘(𝐸‘𝑗)) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))))
138	137	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ) → (𝑗 ∈ 𝑍 → (𝑥 < (𝑀‘(𝐸‘𝑗)) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))))
139	117, 118, 138	rexlimd 3248	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))))
140	110, 139	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
141	108, 140	ralrimia 3429	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
142		xrpnf 42997	. . . . . . 7 ⊢ ((𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^* → ((𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = +∞ ↔ ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))))
143	124, 142	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = +∞ ↔ ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))))
144	143	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ((𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = +∞ ↔ ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))))
145	141, 144	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = +∞)
146	105, 145	breqtrrd 5107	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
147	47, 146	syldan 591	. 2 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
148	25, 147	pm2.61dan 810	1 ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 Vcvv 3431 ⊆ wss 3892 ∪ ciun 4930 class class class wbr 5079 ↦ cmpt 5162 dom cdm 5590 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ωcom 7706 ≼ cdom 8714 ℝcr 10871 1c1 10873 + caddc 10875 +∞cpnf 11007 ℝ^*cxr 11009 < clt 11010 ≤ cle 11011 ℤcz 12319 ℤ_≥cuz 12581 ..^cfzo 13381 ~~>*clsxlim 43330 Meascmea 43958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-disj 5045 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-oadd 8292 df-omul 8293 df-er 8481 df-map 8600 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-acn 9701 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ioc 13083 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-rlim 15196 df-sum 15396 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-rest 17131 df-topn 17132 df-topgen 17152 df-ordt 17210 df-ps 18282 df-tsr 18283 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-lm 22378 df-xms 23471 df-ms 23472 df-xlim 43331 df-salg 43821 df-sumge0 43872 df-mea 43959

This theorem is referenced by: meaiuninc3 43994

W3C validator