meascnbl - Mathbox for Thierry Arnoux

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Theorem meascnbl 34183

Description: A measure is continuous from below. Cf. volsup 25610. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)

**Hypotheses**
Ref	Expression
meascnbl.0	⊢ 𝐽 = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
meascnbl.1	⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
meascnbl.2	⊢ (𝜑 → 𝐹:ℕ⟶𝑆)
meascnbl.3	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))

**Assertion**
Ref	Expression
meascnbl	⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝_𝑡‘𝐽)(𝑀‘∪ ran 𝐹))

Distinct variable groups: 𝑛,𝐹 𝑛,𝐽 𝑛,𝑀 𝑆,𝑛 𝜑,𝑛

Proof of Theorem meascnbl Dummy variables 𝑖 𝑘 𝑜 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		meascnbl.0	. . 3 ⊢ 𝐽 = (TopOpen‘(ℝ^*_𝑠 ↾_s (0[,]+∞)))
2		meascnbl.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆))
4		measbase 34161	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
5	2, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ∪ ran sigAlgebra)
7		meascnbl.2	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶𝑆)
8	7	ffvelcdmda 7118	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑆)
9		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑)
10		fzossnn 13765	. . . . . . . . 9 ⊢ (1..^𝑛) ⊆ ℕ
11		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ (1..^𝑛))
12	10, 11	sselid 4006	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ ℕ)
13	7	ffvelcdmda 7118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑆)
14	9, 12, 13	syl2anc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → (𝐹‘𝑘) ∈ 𝑆)
15	14	ralrimiva 3152	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
16		sigaclfu2 34085	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
17	6, 15, 16	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
18		difelsiga 34097	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ (𝐹‘𝑛) ∈ 𝑆 ∧ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) ∈ 𝑆)
19	6, 8, 17, 18	syl3anc 1371	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) ∈ 𝑆)
20		measvxrge0 34169	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) ∈ 𝑆) → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) ∈ (0[,]+∞))
21	3, 19, 20	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) ∈ (0[,]+∞))
22		fveq2 6920	. . . . 5 ⊢ (𝑛 = 𝑜 → (𝐹‘𝑛) = (𝐹‘𝑜))
23		oveq2 7456	. . . . . 6 ⊢ (𝑛 = 𝑜 → (1..^𝑛) = (1..^𝑜))
24	23	iuneq1d 5042	. . . . 5 ⊢ (𝑛 = 𝑜 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘))
25	22, 24	difeq12d 4150	. . . 4 ⊢ (𝑛 = 𝑜 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘)))
26	25	fveq2d 6924	. . 3 ⊢ (𝑛 = 𝑜 → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘))))
27		fveq2 6920	. . . . 5 ⊢ (𝑛 = 𝑝 → (𝐹‘𝑛) = (𝐹‘𝑝))
28		oveq2 7456	. . . . . 6 ⊢ (𝑛 = 𝑝 → (1..^𝑛) = (1..^𝑝))
29	28	iuneq1d 5042	. . . . 5 ⊢ (𝑛 = 𝑝 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘))
30	27, 29	difeq12d 4150	. . . 4 ⊢ (𝑛 = 𝑝 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑝) ∖ ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘)))
31	30	fveq2d 6924	. . 3 ⊢ (𝑛 = 𝑝 → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘((𝐹‘𝑝) ∖ ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘))))
32	1, 21, 26, 31	esumcvg2 34051	. 2 ⊢ (𝜑 → (𝑖 ∈ ℕ ↦ Σ^𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))(⇝_𝑡‘𝐽)Σ^𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))
33		measfrge0 34167	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
34	2, 33	syl 17	. . . 4 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
35		fcompt 7167	. . . 4 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝐹:ℕ⟶𝑆) → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖))))
36	34, 7, 35	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖))))
37		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑛(𝐹‘𝑘)
38		fveq2 6920	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
39		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
40	39	nnzd 12666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
41		fzval3 13785	. . . . . . . 8 ⊢ (𝑖 ∈ ℤ → (1...𝑖) = (1..^(𝑖 + 1)))
42	40, 41	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1...𝑖) = (1..^(𝑖 + 1)))
43	42	olcd 873	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1...𝑖) = ℕ ∨ (1...𝑖) = (1..^(𝑖 + 1))))
44	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆))
45		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝜑)
46		fzossnn 13765	. . . . . . . 8 ⊢ (1..^(𝑖 + 1)) ⊆ ℕ
47	42	eleq2d 2830	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑛 ∈ (1...𝑖) ↔ 𝑛 ∈ (1..^(𝑖 + 1))))
48	47	biimpa 476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝑛 ∈ (1..^(𝑖 + 1)))
49	46, 48	sselid 4006	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝑛 ∈ ℕ)
50	45, 49, 8	syl2anc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → (𝐹‘𝑛) ∈ 𝑆)
51	37, 38, 43, 44, 50	measiuns 34181	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑀‘∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛)) = Σ^*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))
52	7	ffnd 6748	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ)
53		meascnbl.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
54	52, 53	iuninc 32583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))
55	54	fveq2d 6924	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑀‘∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛)) = (𝑀‘(𝐹‘𝑖)))
56	51, 55	eqtr3d 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → Σ^*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘(𝐹‘𝑖)))
57	56	mpteq2dva 5266	. . 3 ⊢ (𝜑 → (𝑖 ∈ ℕ ↦ Σ^*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)))) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖))))
58	36, 57	eqtr4d 2783	. 2 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ Σ^*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)))))
59	8	ralrimiva 3152	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑆)
60		dfiun2g 5053	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑆 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (𝐹‘𝑛)})
61	59, 60	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (𝐹‘𝑛)})
62		fnrnfv 6981	. . . . . . 7 ⊢ (𝐹 Fn ℕ → ran 𝐹 = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (𝐹‘𝑛)})
63	52, 62	syl 17	. . . . . 6 ⊢ (𝜑 → ran 𝐹 = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (𝐹‘𝑛)})
64	63	unieqd 4944	. . . . 5 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (𝐹‘𝑛)})
65	61, 64	eqtr4d 2783	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
66	65	fveq2d 6924	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑀‘∪ ran 𝐹))
67		eqidd 2741	. . . . 5 ⊢ (𝜑 → ℕ = ℕ)
68	67	orcd 872	. . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^(𝑖 + 1))))
69	37, 38, 68, 2, 8	measiuns 34181	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = Σ^*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))
70	66, 69	eqtr3d 2782	. 2 ⊢ (𝜑 → (𝑀‘∪ ran 𝐹) = Σ^*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))
71	32, 58, 70	3brtr4d 5198	1 ⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝_𝑡‘𝐽)(𝑀‘∪ ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {cab 2717 ∀wral 3067 ∃wrex 3076 ∖ cdif 3973 ⊆ wss 3976 ∪ cuni 4931 ∪ ciun 5015 class class class wbr 5166 ↦ cmpt 5249 ran crn 5701 ∘ ccom 5704 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 0cc0 11184 1c1 11185 + caddc 11187 +∞cpnf 11321 ℕcn 12293 ℤcz 12639 [,]cicc 13410 ...cfz 13567 ..^cfzo 13711 ↾_s cress 17287 TopOpenctopn 17481 ℝ^*_𝑠cxrs 17560 ⇝_𝑡clm 23255 Σ^*cesum 33991 sigAlgebracsiga 34072 measurescmeas 34159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-ac2 10532 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-ac 10185 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-ordt 17561 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-ps 18636 df-tsr 18637 df-plusf 18677 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-abv 20832 df-lmod 20882 df-scaf 20883 df-sra 21195 df-rgmod 21196 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-lm 23258 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-tmd 24101 df-tgp 24102 df-tsms 24156 df-trg 24189 df-xms 24351 df-ms 24352 df-tms 24353 df-nm 24616 df-ngp 24617 df-nrg 24619 df-nlm 24620 df-ii 24922 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-esum 33992 df-siga 34073 df-meas 34160

This theorem is referenced by: dstfrvclim1 34442

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