meaiuninc3v - Mathbox for Glauco Siliprandi

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Theorem meaiuninc3v 47226

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 47223 and meaiuninc2 47224 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 485	. . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
6		eqid 2763	. . . . . 6 ⊢ dom 𝑀 = dom 𝑀
7		meaiuninc3v.e	. . . . . . 7 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
8	7	ffvelcdmda 7079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
9	5, 6, 8	meaxrcl 47203	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
10		meaiuninc3v.s	. . . . 5 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
11	9, 10	fmptd 7109	. . . 4 ⊢ (𝜑 → 𝑆:𝑍⟶ℝ^*)
12	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆:𝑍⟶ℝ^*)
13		nfv 1944	. . . . 5 ⊢ Ⅎ𝑛𝜑
14		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑛ℝ
15		nfra1 3289	. . . . . 6 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥
16	14, 15	nfrexw 3313	. . . . 5 ⊢ Ⅎ𝑛∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥
17	13, 16	nfan 1929	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
18		nfcv 2925	. . . 4 ⊢ Ⅎ𝑛𝐸
19	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑀 ∈ Meas)
20	7	adantr 485	. . . 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 727	. . . 4 ⊢ (((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
23		simpr 489	. . . 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 47225	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
25	2, 3, 12, 24	climxlim2 46588	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
26		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
27		2fveq3 6886	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑛 → (𝑀‘(𝐸‘𝑗)) = (𝑀‘(𝐸‘𝑛)))
28	27	breq2d 5121	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ 𝑥 < (𝑀‘(𝐸‘𝑛))))
29	28	cbvrexvw 3244	. . . . . . . . . 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 11259	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
32	31	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ℝ^*)
33	9	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
34	32, 33	xrltnled 11281	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑥 < (𝑀‘(𝐸‘𝑛)) ↔ ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
35	34	rexbidva 3187	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑛)) ↔ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
36	30, 35	bitrd 282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
37	36	ralbidva 3186	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
38		rexnal 3117	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
39	38	ralbii 3111	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
40		ralnex 3091	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
41	39, 40	bitri 278	. . . . . . . 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 282	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
44	43	adantr 485	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
46		simpr 489	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
47	45, 46	syldan 602	. . 3 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
48		simp-4r 795	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ)
49	48, 31	syl 18	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ^*)
50		simp-4l 794	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝜑)
51	3	uztrn2 12885	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
52	51	ad4ant24 766	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
53	11	ffvelcdmda 7079	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) ∈ ℝ^*)
54	50, 52, 53	syl2anc 595	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) ∈ ℝ^*)
55		eleq1w 2846	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑛 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
56	55	anbi2d 641	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
57		2fveq3 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑗)))
58	57	eleq1d 2848	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝑀‘(𝐸‘𝑛)) ∈ ℝ^* ↔ (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*))
59	56, 58	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^)))
60	59, 9	chvarvv 2019	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
61	60	ad5ant13 768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
62		simplr 780	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 < (𝑀‘(𝐸‘𝑗)))
63	4	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑀 ∈ Meas)
64	7	ffvelcdmda 7079	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ dom 𝑀)
65	64	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑗) ∈ dom 𝑀)
66		simp1 1154	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝜑)
67	51	3adant1 1148	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
68	66, 67, 8	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑛) ∈ dom 𝑀)
69		simp3 1156	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ (ℤ_≥‘𝑗))
70		simpll 778	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝜑)
71	3	uzssd3 46168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
72	71	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → (ℤ_≥‘𝑗) ⊆ 𝑍)
73		elfzouz 13697	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (𝑗..^𝑛) → 𝑘 ∈ (ℤ_≥‘𝑗))
74	73	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ (ℤ_≥‘𝑗))
75	72, 74	sseldd 3938	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ 𝑍)
76	75	adantll 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ 𝑍)
77		eleq1w 2846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍))
78	77	anbi2d 641	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍)))
79		fveq2 6881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘))
80		fvoveq1 7433	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1)))
81	79, 80	sseq12d 3970	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))))
82	78, 81	imbi12d 347	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))))
83	82, 21	chvarvv 2019	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
84	70, 76, 83	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
85	84	3adantl3 1187	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) ∧ 𝑘 ∈ (𝑗..^𝑛)) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
86	69, 85	ssinc 45833	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑗) ⊆ (𝐸‘𝑛))
87	63, 6, 65, 68, 86	meassle 47205	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘(𝐸‘𝑛)))
88		fvexd 6896	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑛)) ∈ V)
89	10	fvmpt2 7001	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝑀‘(𝐸‘𝑛)) ∈ V) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
90	51, 88, 89	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
91	90	3adant1 1148	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
92	87, 91	breqtrrd 5139	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑆‘𝑛))
93	92	ad5ant135 1394	. . . . . . . . . . . 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 13190	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 < (𝑆‘𝑛))
95	49, 54, 94	xrltled 13179	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 ≤ (𝑆‘𝑛))
96	95	ralrimiva 3157	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛))
97	96	ex 417	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝑥 < (𝑀‘(𝐸‘𝑗)) → ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
98	97	reximdva 3178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
99	98	ralimdva 3177	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
100	99	imp 411	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛))
101		nfmpt1 5210	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
102	10, 101	nfcxfr 2923	. . . . . . 7 ⊢ Ⅎ𝑛𝑆
103	102, 1, 3, 11	xlimpnf 46584	. . . . . 6 ⊢ (𝜑 → (𝑆~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
104	103	adantr 485	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑆~~>*+∞)
106		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
107		nfra1 3289	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
108	106, 107	nfan 1929	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
109		rspa 3254	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
110	109	adantll 726	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
111		nfv 1944	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝜑
112		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ℝ
113		nfre1 3290	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
114	112, 113	nfralw 3312	. . . . . . . . . 10 ⊢ Ⅎ𝑗∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
115	111, 114	nfan 1929	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
116		nfv 1944	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
117	115, 116	nfan 1929	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ)
118		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
119	31	ad3antlr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 ∈ ℝ^*)
120	4, 6	dmmeasal 47194	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
121	3	uzct 45811	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ≼ ω
122	121	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ≼ ω)
123	120, 122, 8	saliuncl 47065	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ dom 𝑀)
124	4, 6, 123	meaxrcl 47203	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
125	124	ad3antrrr 742	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
126	60	ad4ant13 763	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
127		simpr 489	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 < (𝑀‘(𝐸‘𝑗)))
128	4	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑀 ∈ Meas)
129	123	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ dom 𝑀)
130		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝐸‘𝑛) = (𝐸‘𝑗))
131	130	ssiun2s 5013	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (𝐸‘𝑗) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
132	131	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
133	128, 6, 64, 129, 132	meassle 47205	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
134	133	ad4ant13 763	. . . . . . . . . . . 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 13190	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 < (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
136	119, 125, 135	xrltled 13179	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
137	136	exp31 424	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑗 ∈ 𝑍 → (𝑥 < (𝑀‘(𝐸‘𝑗)) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))))
138	137	adantlr 727	. . . . . . . 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 3272	. . . . . . 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 16	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
141	108, 140	ralrimia 3264	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
142		xrpnf 46227	. . . . . . 7 ⊢ ((𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^* → ((𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = +∞ ↔ ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))))
143	124, 142	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = +∞ ↔ ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))))
144	143	adantr 485	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = +∞)
146	105, 145	breqtrrd 5139	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
147	47, 146	syldan 602	. 2 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
148	25, 147	pm2.61dan 824	1 ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ⊆ wss 3905 ∪ ciun 4956 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5661 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ωcom 7858 ≼ cdom 8937 ℝcr 11103 1c1 11105 + caddc 11107 +∞cpnf 11244 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 ℤcz 12595 ℤ_≥cuz 12866 ..^cfzo 13687 ~~>*clsxlim 46560 Meascmea 47191

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-disj 5077 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-omul 8454 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9930 df-acn 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13380 df-ioc 13381 df-ico 13382 df-icc 13383 df-fz 13540 df-fzo 13688 df-fl 13830 df-seq 14043 df-exp 14103 df-hash 14372 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-clim 15544 df-rlim 15545 df-sum 15743 df-struct 17211 df-slot 17246 df-ndx 17258 df-base 17274 df-plusg 17327 df-mulr 17328 df-starv 17329 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-rest 17479 df-topn 17480 df-topgen 17500 df-ordt 17559 df-ps 18626 df-tsr 18627 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-cnfld 21532 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-lm 23395 df-xms 24486 df-ms 24487 df-xlim 46561 df-salg 47051 df-sumge0 47105 df-mea 47192

This theorem is used by: meaiuninc3 47227

W3C validator