meaiuninc3v - Mathbox for Glauco Siliprandi

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Theorem meaiuninc3v 46912

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 46909 and meaiuninc2 46910 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 480	. . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑀 ∈ Meas)
6		eqid 2736	. . . . . 6 ⊢ dom 𝑀 = dom 𝑀
7		meaiuninc3v.e	. . . . . . 7 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
8	7	ffvelcdmda 7036	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
9	5, 6, 8	meaxrcl 46889	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
10		meaiuninc3v.s	. . . . 5 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
11	9, 10	fmptd 7066	. . . 4 ⊢ (𝜑 → 𝑆:𝑍⟶ℝ^*)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆:𝑍⟶ℝ^*)
13		nfv 1916	. . . . 5 ⊢ Ⅎ𝑛𝜑
14		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑛ℝ
15		nfra1 3261	. . . . . 6 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥
16	14, 15	nfrexw 3285	. . . . 5 ⊢ Ⅎ𝑛∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥
17	13, 16	nfan 1901	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
18		nfcv 2898	. . . 4 ⊢ Ⅎ𝑛𝐸
19	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑀 ∈ Meas)
20	7	adantr 480	. . . 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 716	. . . 4 ⊢ (((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)))
23		simpr 484	. . . 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 46911	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
25	2, 3, 12, 24	climxlim2 46274	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
26		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
27		2fveq3 6845	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑛 → (𝑀‘(𝐸‘𝑗)) = (𝑀‘(𝐸‘𝑛)))
28	27	breq2d 5097	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ 𝑥 < (𝑀‘(𝐸‘𝑛))))
29	28	cbvrexvw 3216	. . . . . . . . . 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 11191	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
32	31	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ℝ^*)
33	9	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
34	32, 33	xrltnled 11213	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑥 < (𝑀‘(𝐸‘𝑛)) ↔ ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
35	34	rexbidva 3159	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑛 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑛)) ↔ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
36	30, 35	bitrd 279	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
37	36	ralbidva 3158	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
38		rexnal 3089	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
39	38	ralbii 3083	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
40		ralnex 3063	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
41	39, 40	bitri 275	. . . . . . . 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 279	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
44	43	adantr 480	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
46		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
47	45, 46	syldan 592	. . 3 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
48		simp-4r 784	. . . . . . . . . . . 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 783	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝜑)
51	3	uztrn2 12807	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
52	51	ad4ant24 755	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
53	11	ffvelcdmda 7036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) ∈ ℝ^*)
54	50, 52, 53	syl2anc 585	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) ∈ ℝ^*)
55		eleq1w 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑛 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
56	55	anbi2d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
57		2fveq3 6845	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑗)))
58	57	eleq1d 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝑀‘(𝐸‘𝑛)) ∈ ℝ^* ↔ (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*))
59	56, 58	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑗 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^)))
60	59, 9	chvarvv 1991	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
61	60	ad5ant13 757	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
62		simplr 769	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 < (𝑀‘(𝐸‘𝑗)))
63	4	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑀 ∈ Meas)
64	7	ffvelcdmda 7036	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ dom 𝑀)
65	64	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑗) ∈ dom 𝑀)
66		simp1 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝜑)
67	51	3adant1 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
68	66, 67, 8	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑛) ∈ dom 𝑀)
69		simp3 1139	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ (ℤ_≥‘𝑗))
70		simpll 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝜑)
71	3	uzssd3 45854	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
72	71	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → (ℤ_≥‘𝑗) ⊆ 𝑍)
73		elfzouz 13618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (𝑗..^𝑛) → 𝑘 ∈ (ℤ_≥‘𝑗))
74	73	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ (ℤ_≥‘𝑗))
75	72, 74	sseldd 3922	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ 𝑍)
76	75	adantll 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ 𝑍)
77		eleq1w 2819	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍))
78	77	anbi2d 631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍)))
79		fveq2 6840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘))
80		fvoveq1 7390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1)))
81	79, 80	sseq12d 3955	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))))
82	78, 81	imbi12d 344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))))
83	82, 21	chvarvv 1991	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
84	70, 76, 83	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
85	84	3adantl3 1170	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) ∧ 𝑘 ∈ (𝑗..^𝑛)) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
86	69, 85	ssinc 45517	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑗) ⊆ (𝐸‘𝑛))
87	63, 6, 65, 68, 86	meassle 46891	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘(𝐸‘𝑛)))
88		fvexd 6855	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑛)) ∈ V)
89	10	fvmpt2 6959	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝑀‘(𝐸‘𝑛)) ∈ V) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
90	51, 88, 89	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
91	90	3adant1 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
92	87, 91	breqtrrd 5113	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑆‘𝑛))
93	92	ad5ant135 1371	. . . . . . . . . . . 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 13112	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 < (𝑆‘𝑛))
95	49, 54, 94	xrltled 13101	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 ≤ (𝑆‘𝑛))
96	95	ralrimiva 3129	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛))
97	96	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝑥 < (𝑀‘(𝐸‘𝑗)) → ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
98	97	reximdva 3150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
99	98	ralimdva 3149	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
100	99	imp 406	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛))
101		nfmpt1 5184	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
102	10, 101	nfcxfr 2896	. . . . . . 7 ⊢ Ⅎ𝑛𝑆
103	102, 1, 3, 11	xlimpnf 46270	. . . . . 6 ⊢ (𝜑 → (𝑆~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
104	103	adantr 480	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑆~~>*+∞)
106		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
107		nfra1 3261	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
108	106, 107	nfan 1901	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
109		rspa 3226	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
110	109	adantll 715	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
111		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝜑
112		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ℝ
113		nfre1 3262	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
114	112, 113	nfralw 3284	. . . . . . . . . 10 ⊢ Ⅎ𝑗∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
115	111, 114	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
116		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
117	115, 116	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ)
118		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
119	31	ad3antlr 732	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 ∈ ℝ^*)
120	4, 6	dmmeasal 46880	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
121	3	uzct 45494	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ≼ ω
122	121	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ≼ ω)
123	120, 122, 8	saliuncl 46751	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ dom 𝑀)
124	4, 6, 123	meaxrcl 46889	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
125	124	ad3antrrr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
126	60	ad4ant13 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
127		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 < (𝑀‘(𝐸‘𝑗)))
128	4	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑀 ∈ Meas)
129	123	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ dom 𝑀)
130		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝐸‘𝑛) = (𝐸‘𝑗))
131	130	ssiun2s 4991	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (𝐸‘𝑗) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
132	131	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ⊆ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
133	128, 6, 64, 129, 132	meassle 46891	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
134	133	ad4ant13 752	. . . . . . . . . . . 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 13112	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 < (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
136	119, 125, 135	xrltled 13101	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
137	136	exp31 419	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑗 ∈ 𝑍 → (𝑥 < (𝑀‘(𝐸‘𝑗)) → 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))))
138	137	adantlr 716	. . . . . . . 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 3244	. . . . . . 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 3236	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
142		xrpnf 45913	. . . . . . 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 480	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = +∞)
146	105, 145	breqtrrd 5113	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
147	47, 146	syldan 592	. 2 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
148	25, 147	pm2.61dan 813	1 ⊢ (𝜑 → 𝑆~~>*(𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ⊆ wss 3889 ∪ ciun 4933 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ωcom 7817 ≼ cdom 8891 ℝcr 11037 1c1 11039 + caddc 11041 +∞cpnf 11176 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 ℤcz 12524 ℤ_≥cuz 12788 ..^cfzo 13608 ~~>*clsxlim 46246 Meascmea 46877

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-ordt 17465 df-ps 18532 df-tsr 18533 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-lm 23194 df-xms 24285 df-ms 24286 df-xlim 46247 df-salg 46737 df-sumge0 46791 df-mea 46878

This theorem is referenced by: meaiuninc3 46913

W3C validator