meaiuninc3v - Mathbox for Glauco Siliprandi

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Theorem meaiuninc3v 46489

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 46486 and meaiuninc2 46487 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 2730	. . . . . 6 ⊢ dom 𝑀 = dom 𝑀
7		meaiuninc3v.e	. . . . . . 7 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀)
8	7	ffvelcdmda 7059	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑀)
9	5, 6, 8	meaxrcl 46466	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
10		meaiuninc3v.s	. . . . 5 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
11	9, 10	fmptd 7089	. . . 4 ⊢ (𝜑 → 𝑆:𝑍⟶ℝ^*)
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆:𝑍⟶ℝ^*)
13		nfv 1914	. . . . 5 ⊢ Ⅎ𝑛𝜑
14		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑛ℝ
15		nfra1 3262	. . . . . 6 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥
16	14, 15	nfrexw 3289	. . . . 5 ⊢ Ⅎ𝑛∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥
17	13, 16	nfan 1899	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
18		nfcv 2892	. . . 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 715	. . . 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 46488	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
25	2, 3, 12, 24	climxlim2 45851	. 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 6866	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑛 → (𝑀‘(𝐸‘𝑗)) = (𝑀‘(𝐸‘𝑛)))
28	27	breq2d 5122	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ 𝑥 < (𝑀‘(𝐸‘𝑛))))
29	28	cbvrexvw 3217	. . . . . . . . . 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 11227	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
32	31	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ℝ^*)
33	9	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑀‘(𝐸‘𝑛)) ∈ ℝ^*)
34	32, 33	xrltnled 45366	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑥 < (𝑀‘(𝐸‘𝑛)) ↔ ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
35	34	rexbidva 3156	. . . . . . . . 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 3155	. . . . . . 7 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ↔ ∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥))
38		rexnal 3083	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
39	38	ralbii 3076	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ∃𝑛 ∈ 𝑍 ¬ (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ¬ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥)
40		ralnex 3056	. . . . . . . . 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 591	. . 3 ⊢ ((𝜑 ∧ ¬ ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
48		simp-4r 783	. . . . . . . . . . . 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 782	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝜑)
51	3	uztrn2 12819	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
52	51	ad4ant24 754	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
53	11	ffvelcdmda 7059	. . . . . . . . . . . 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 2812	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑛 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
56	55	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑗 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
57		2fveq3 6866	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑗)))
58	57	eleq1d 2814	. . . . . . . . . . . . . . 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 1989	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
61	60	ad5ant13 756	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ∈ ℝ^*)
62		simplr 768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 < (𝑀‘(𝐸‘𝑗)))
63	4	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑀 ∈ Meas)
64	7	ffvelcdmda 7059	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ dom 𝑀)
65	64	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑗) ∈ dom 𝑀)
66		simp1 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝜑)
67	51	3adant1 1130	. . . . . . . . . . . . . . . 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 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ (ℤ_≥‘𝑗))
70		simpll 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝜑)
71	3	uzssd3 45429	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
72	71	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → (ℤ_≥‘𝑗) ⊆ 𝑍)
73		elfzouz 13631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (𝑗..^𝑛) → 𝑘 ∈ (ℤ_≥‘𝑗))
74	73	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ (ℤ_≥‘𝑗))
75	72, 74	sseldd 3950	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ 𝑍)
76	75	adantll 714	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (𝑗..^𝑛)) → 𝑘 ∈ 𝑍)
77		eleq1w 2812	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍))
78	77	anbi2d 630	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍)))
79		fveq2 6861	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘))
80		fvoveq1 7413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1)))
81	79, 80	sseq12d 3983	. . . . . . . . . . . . . . . . . . . 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 1989	. . . . . . . . . . . . . . . . . 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 1169	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) ∧ 𝑘 ∈ (𝑗..^𝑛)) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))
86	69, 85	ssinc 45088	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝐸‘𝑗) ⊆ (𝐸‘𝑛))
87	63, 6, 65, 68, 86	meassle 46468	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘(𝐸‘𝑛)))
88		fvexd 6876	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑛)) ∈ V)
89	10	fvmpt2 6982	. . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑆‘𝑛) = (𝑀‘(𝐸‘𝑛)))
92	87, 91	breqtrrd 5138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑆‘𝑛))
93	92	ad5ant135 1370	. . . . . . . . . . . 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 13128	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 < (𝑆‘𝑛))
95	49, 54, 94	xrltled 13117	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑥 ≤ (𝑆‘𝑛))
96	95	ralrimiva 3126	. . . . . . . . 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 3147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)𝑥 ≤ (𝑆‘𝑛)))
99	98	ralimdva 3146	. . . . . 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 5209	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))
102	10, 101	nfcxfr 2890	. . . . . . 7 ⊢ Ⅎ𝑛𝑆
103	102, 1, 3, 11	xlimpnf 45847	. . . . . 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 1914	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
107		nfra1 3262	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
108	106, 107	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
109		rspa 3227	. . . . . . . 8 ⊢ ((∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
110	109	adantll 714	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ) → ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
111		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝜑
112		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑗ℝ
113		nfre1 3263	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
114	112, 113	nfralw 3287	. . . . . . . . . 10 ⊢ Ⅎ𝑗∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))
115	111, 114	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗)))
116		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑥 ∈ ℝ
117	115, 116	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑗((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) ∧ 𝑥 ∈ ℝ)
118		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))
119	31	ad3antlr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 ∈ ℝ^*)
120	4, 6	dmmeasal 46457	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
121	3	uzct 45064	. . . . . . . . . . . . . . 15 ⊢ 𝑍 ≼ ω
122	121	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ≼ ω)
123	120, 122, 8	saliuncl 46328	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ dom 𝑀)
124	4, 6, 123	meaxrcl 46466	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
125	124	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
126	60	ad4ant13 751	. . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝐸‘𝑛) = (𝐸‘𝑗))
131	130	ssiun2s 5015	. . . . . . . . . . . . . . 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 46468	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀‘(𝐸‘𝑗)) ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
134	133	ad4ant13 751	. . . . . . . . . . . 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 13128	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ 𝑥 < (𝑀‘(𝐸‘𝑗))) → 𝑥 < (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
136	119, 125, 135	xrltled 13117	. . . . . . . . . 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 715	. . . . . . . 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 3245	. . . . . . 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 3237	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 𝑥 < (𝑀‘(𝐸‘𝑗))) → ∀𝑥 ∈ ℝ 𝑥 ≤ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)))
142		xrpnf 45488	. . . . . . 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 5138	. . 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 812	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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ⊆ wss 3917 ∪ ciun 4958 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ωcom 7845 ≼ cdom 8919 ℝcr 11074 1c1 11076 + caddc 11078 +∞cpnf 11212 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 ℤcz 12536 ℤ_≥cuz 12800 ..^cfzo 13622 ~~>*clsxlim 45823 Meascmea 46454

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-acn 9902 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17392 df-topn 17393 df-topgen 17413 df-ordt 17471 df-ps 18532 df-tsr 18533 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-lm 23123 df-xms 24215 df-ms 24216 df-xlim 45824 df-salg 46314 df-sumge0 46368 df-mea 46455

This theorem is referenced by: meaiuninc3 46490

W3C validator