meascnbl - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > meascnbl		Structured version Visualization version GIF version

Theorem meascnbl 34415

Description: A measure is continuous from below. Cf. volsup 25545. (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 482	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆))
4		measbase 34393	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
5	2, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
6	5	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ∪ ran sigAlgebra)
7		meascnbl.2	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶𝑆)
8	7	ffvelcdmda 7029	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑆)
9		simpll 773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑)
10		fzossnn 13661	. . . . . . . . 9 ⊢ (1..^𝑛) ⊆ ℕ
11		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ (1..^𝑛))
12	10, 11	sselid 3915	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ ℕ)
13	7	ffvelcdmda 7029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑆)
14	9, 12, 13	syl2anc 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → (𝐹‘𝑘) ∈ 𝑆)
15	14	ralrimiva 3133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
16		sigaclfu2 34317	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
17	6, 15, 16	syl2anc 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆)
18		difelsiga 34329	. . . . 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 1380	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) ∈ 𝑆)
20		measvxrge0 34401	. . . 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 591	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) ∈ (0[,]+∞))
22		fveq2 6831	. . . . 5 ⊢ (𝑛 = 𝑜 → (𝐹‘𝑛) = (𝐹‘𝑜))
23		oveq2 7368	. . . . . 6 ⊢ (𝑛 = 𝑜 → (1..^𝑛) = (1..^𝑜))
24	23	iuneq1d 4952	. . . . 5 ⊢ (𝑛 = 𝑜 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘))
25	22, 24	difeq12d 4061	. . . 4 ⊢ (𝑛 = 𝑜 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘)))
26	25	fveq2d 6835	. . 3 ⊢ (𝑛 = 𝑜 → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘))))
27		fveq2 6831	. . . . 5 ⊢ (𝑛 = 𝑝 → (𝐹‘𝑛) = (𝐹‘𝑝))
28		oveq2 7368	. . . . . 6 ⊢ (𝑛 = 𝑝 → (1..^𝑛) = (1..^𝑝))
29	28	iuneq1d 4952	. . . . 5 ⊢ (𝑛 = 𝑝 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘))
30	27, 29	difeq12d 4061	. . . 4 ⊢ (𝑛 = 𝑝 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑝) ∖ ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘)))
31	30	fveq2d 6835	. . 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 34283	. 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 34399	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
34	2, 33	syl 17	. . . 4 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
35		fcompt 7079	. . . 4 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝐹:ℕ⟶𝑆) → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖))))
36	34, 7, 35	syl2anc 591	. . 3 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖))))
37		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑛(𝐹‘𝑘)
38		fveq2 6831	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
39		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
40	39	nnzd 12545	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ)
41		fzval3 13684	. . . . . . . 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 881	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1...𝑖) = ℕ ∨ (1...𝑖) = (1..^(𝑖 + 1))))
44	2	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆))
45		simpll 773	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝜑)
46		fzossnn 13661	. . . . . . . 8 ⊢ (1..^(𝑖 + 1)) ⊆ ℕ
47	42	eleq2d 2827	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑛 ∈ (1...𝑖) ↔ 𝑛 ∈ (1..^(𝑖 + 1))))
48	47	biimpa 478	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝑛 ∈ (1..^(𝑖 + 1)))
49	46, 48	sselid 3915	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝑛 ∈ ℕ)
50	45, 49, 8	syl2anc 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → (𝐹‘𝑛) ∈ 𝑆)
51	37, 38, 43, 44, 50	measiuns 34413	. . . . 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 6660	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ)
53		meascnbl.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))
54	52, 53	iuninc 32653	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖))
55	54	fveq2d 6835	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑀‘∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛)) = (𝑀‘(𝐹‘𝑖)))
56	51, 55	eqtr3d 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → Σ^*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘(𝐹‘𝑖)))
57	56	mpteq2dva 5168	. . 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 2779	. 2 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ Σ^*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)))))
59	8	ralrimiva 3133	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑆)
60		dfiun2g 4962	. . . . . 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 6890	. . . . . . 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 4854	. . . . 5 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (𝐹‘𝑛)})
65	61, 64	eqtr4d 2779	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹)
66	65	fveq2d 6835	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑀‘∪ ran 𝐹))
67		eqidd 2742	. . . . 5 ⊢ (𝜑 → ℕ = ℕ)
68	67	orcd 880	. . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^(𝑖 + 1))))
69	37, 38, 68, 2, 8	measiuns 34413	. . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = Σ^*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))
70	66, 69	eqtr3d 2778	. 2 ⊢ (𝜑 → (𝑀‘∪ ran 𝐹) = Σ^*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))
71	32, 58, 70	3brtr4d 5107	1 ⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝_𝑡‘𝐽)(𝑀‘∪ ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 {cab 2719 ∀wral 3055 ∃wrex 3065 ∖ cdif 3882 ⊆ wss 3885 ∪ cuni 4841 ∪ ciun 4924 class class class wbr 5075 ↦ cmpt 5156 ran crn 5622 ∘ ccom 5625 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 0cc0 11033 1c1 11034 + caddc 11036 +∞cpnf 11171 ℕcn 12169 ℤcz 12519 [,]cicc 13296 ...cfz 13456 ..^cfzo 13603 ↾_s cress 17195 TopOpenctopn 17379 ℝ^*_𝑠cxrs 17459 ⇝_𝑡clm 23213 Σ^*cesum 34223 sigAlgebracsiga 34304 measurescmeas 34391

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-ac2 10380 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-disj 5043 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9820 df-card 9858 df-acn 9861 df-ac 10033 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-sin 16029 df-cos 16030 df-pi 16032 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-ordt 17460 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-ps 18527 df-tsr 18528 df-plusf 18602 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-subrng 20522 df-subrg 20546 df-abv 20785 df-lmod 20856 df-scaf 20857 df-sra 21167 df-rgmod 21168 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-fbas 21348 df-fg 21349 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-lm 23216 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-tmd 24059 df-tgp 24060 df-tsms 24114 df-trg 24147 df-xms 24307 df-ms 24308 df-tms 24309 df-nm 24569 df-ngp 24570 df-nrg 24572 df-nlm 24573 df-ii 24866 df-cncf 24867 df-limc 25855 df-dv 25856 df-log 26542 df-esum 34224 df-siga 34305 df-meas 34392

This theorem is referenced by: dstfrvclim1 34674

W3C validator