meascnbl - Mathbox for Thierry Arnoux

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Theorem meascnbl 34325

Description: A measure is continuous from below. Cf. volsup 25511. (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 34303	. . . . . . 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 7027	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑆)
9		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑)
10		fzossnn 13625	. . . . . . . . 9 ⊢ (1..^𝑛) ⊆ ℕ
11		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ (1..^𝑛))
12	10, 11	sselid 3929	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ ℕ)
13	7	ffvelcdmda 7027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑆)
14	9, 12, 13	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → (𝐹‘𝑘) ∈ 𝑆)
15	14	ralrimiva 3126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
16		sigaclfu2 34227	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
17	6, 15, 16	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
18		difelsiga 34239	. . . . 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 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) ∈ 𝑆)
20		measvxrge0 34311	. . . 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 584	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) ∈ (0[,]+∞))
22		fveq2 6832	. . . . 5 ⊢ (𝑛 = 𝑜 → (𝐹‘𝑛) = (𝐹‘𝑜))
23		oveq2 7364	. . . . . 6 ⊢ (𝑛 = 𝑜 → (1..^𝑛) = (1..^𝑜))
24	23	iuneq1d 4972	. . . . 5 ⊢ (𝑛 = 𝑜 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘))
25	22, 24	difeq12d 4077	. . . 4 ⊢ (𝑛 = 𝑜 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘)))
26	25	fveq2d 6836	. . 3 ⊢ (𝑛 = 𝑜 → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘))))
27		fveq2 6832	. . . . 5 ⊢ (𝑛 = 𝑝 → (𝐹‘𝑛) = (𝐹‘𝑝))
28		oveq2 7364	. . . . . 6 ⊢ (𝑛 = 𝑝 → (1..^𝑛) = (1..^𝑝))
29	28	iuneq1d 4972	. . . . 5 ⊢ (𝑛 = 𝑝 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘))
30	27, 29	difeq12d 4077	. . . 4 ⊢ (𝑛 = 𝑝 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑝) ∖ ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘)))
31	30	fveq2d 6836	. . 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 34193	. 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 34309	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
34	2, 33	syl 17	. . . 4 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
35		fcompt 7076	. . . 4 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝐹:ℕ⟶𝑆) → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖))))
36	34, 7, 35	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖))))
37		nfcv 2896	. . . . . 6 ⊢ Ⅎ𝑛(𝐹‘𝑘)
38		fveq2 6832	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
39		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
40	39	nnzd 12512	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
41		fzval3 13648	. . . . . . . 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 874	. . . . . 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 13625	. . . . . . . 8 ⊢ (1..^(𝑖 + 1)) ⊆ ℕ
47	42	eleq2d 2820	. . . . . . . . 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 3929	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝑛 ∈ ℕ)
50	45, 49, 8	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → (𝐹‘𝑛) ∈ 𝑆)
51	37, 38, 43, 44, 50	measiuns 34323	. . . . 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 6661	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ)
53		meascnbl.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
54	52, 53	iuninc 32584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))
55	54	fveq2d 6836	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑀‘∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛)) = (𝑀‘(𝐹‘𝑖)))
56	51, 55	eqtr3d 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → Σ^*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘(𝐹‘𝑖)))
57	56	mpteq2dva 5189	. . 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 2772	. 2 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ Σ^*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)))))
59	8	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑆)
60		dfiun2g 4983	. . . . . 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 6891	. . . . . . 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 4874	. . . . 5 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (𝐹‘𝑛)})
65	61, 64	eqtr4d 2772	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
66	65	fveq2d 6836	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑀‘∪ ran 𝐹))
67		eqidd 2735	. . . . 5 ⊢ (𝜑 → ℕ = ℕ)
68	67	orcd 873	. . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^(𝑖 + 1))))
69	37, 38, 68, 2, 8	measiuns 34323	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = Σ^*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))
70	66, 69	eqtr3d 2771	. 2 ⊢ (𝜑 → (𝑀‘∪ ran 𝐹) = Σ^*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))
71	32, 58, 70	3brtr4d 5128	1 ⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝_𝑡‘𝐽)(𝑀‘∪ ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {cab 2712 ∀wral 3049 ∃wrex 3058 ∖ cdif 3896 ⊆ wss 3899 ∪ cuni 4861 ∪ ciun 4944 class class class wbr 5096 ↦ cmpt 5177 ran crn 5623 ∘ ccom 5626 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 0cc0 11024 1c1 11025 + caddc 11027 +∞cpnf 11161 ℕcn 12143 ℤcz 12486 [,]cicc 13262 ...cfz 13421 ..^cfzo 13568 ↾_s cress 17155 TopOpenctopn 17339 ℝ^*_𝑠cxrs 17419 ⇝_𝑡clm 23168 Σ^*cesum 34133 sigAlgebracsiga 34214 measurescmeas 34301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-ac2 10371 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103 ax-mulf 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-disj 5064 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-acn 9852 df-ac 10024 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-ef 15988 df-sin 15990 df-cos 15991 df-pi 15993 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-ordt 17420 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-ps 18487 df-tsr 18488 df-plusf 18562 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20477 df-subrg 20501 df-abv 20740 df-lmod 20811 df-scaf 20812 df-sra 21123 df-rgmod 21124 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-lm 23171 df-haus 23257 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-tmd 24014 df-tgp 24015 df-tsms 24069 df-trg 24102 df-xms 24262 df-ms 24263 df-tms 24264 df-nm 24524 df-ngp 24525 df-nrg 24527 df-nlm 24528 df-ii 24824 df-cncf 24825 df-limc 25821 df-dv 25822 df-log 26519 df-esum 34134 df-siga 34215 df-meas 34302

This theorem is referenced by: dstfrvclim1 34584

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