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Mirrors > Home > MPE Home > Th. List > Mathboxes > meascnbl | Structured version Visualization version GIF version |
Description: A measure is continuous from below. Cf. volsup 25605. (Contributed by Thierry Arnoux, 18-Jan-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.) |
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))) |
Ref | Expression |
---|---|
meascnbl | ⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝𝑡‘𝐽)(𝑀‘∪ ran 𝐹)) |
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 34178 | . . . . . . 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 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑆) |
9 | simpll 767 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑) | |
10 | fzossnn 13748 | . . . . . . . . 9 ⊢ (1..^𝑛) ⊆ ℕ | |
11 | simpr 484 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ (1..^𝑛)) | |
12 | 10, 11 | sselid 3993 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ ℕ) |
13 | 7 | ffvelcdmda 7104 | . . . . . . . 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 3144 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) |
16 | sigaclfu2 34102 | . . . . . 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 34114 | . . . . 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 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) ∈ 𝑆) |
20 | measvxrge0 34186 | . . . 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 6907 | . . . . 5 ⊢ (𝑛 = 𝑜 → (𝐹‘𝑛) = (𝐹‘𝑜)) | |
23 | oveq2 7439 | . . . . . 6 ⊢ (𝑛 = 𝑜 → (1..^𝑛) = (1..^𝑜)) | |
24 | 23 | iuneq1d 5024 | . . . . 5 ⊢ (𝑛 = 𝑜 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘)) |
25 | 22, 24 | difeq12d 4137 | . . . 4 ⊢ (𝑛 = 𝑜 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘))) |
26 | 25 | fveq2d 6911 | . . 3 ⊢ (𝑛 = 𝑜 → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘)))) |
27 | fveq2 6907 | . . . . 5 ⊢ (𝑛 = 𝑝 → (𝐹‘𝑛) = (𝐹‘𝑝)) | |
28 | oveq2 7439 | . . . . . 6 ⊢ (𝑛 = 𝑝 → (1..^𝑛) = (1..^𝑝)) | |
29 | 28 | iuneq1d 5024 | . . . . 5 ⊢ (𝑛 = 𝑝 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘)) |
30 | 27, 29 | difeq12d 4137 | . . . 4 ⊢ (𝑛 = 𝑝 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑝) ∖ ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘))) |
31 | 30 | fveq2d 6911 | . . 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 34068 | . 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 34184 | . . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞)) | |
34 | 2, 33 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
35 | fcompt 7153 | . . . 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 2903 | . . . . . 6 ⊢ Ⅎ𝑛(𝐹‘𝑘) | |
38 | fveq2 6907 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) | |
39 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ) | |
40 | 39 | nnzd 12638 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ) |
41 | fzval3 13770 | . . . . . . . 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 767 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝜑) | |
46 | fzossnn 13748 | . . . . . . . 8 ⊢ (1..^(𝑖 + 1)) ⊆ ℕ | |
47 | 42 | eleq2d 2825 | . . . . . . . . 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 3993 | . . . . . . 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 34198 | . . . . 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 6738 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ) |
53 | meascnbl.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) | |
54 | 52, 53 | iuninc 32581 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)) |
55 | 54 | fveq2d 6911 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑀‘∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛)) = (𝑀‘(𝐹‘𝑖))) |
56 | 51, 55 | eqtr3d 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → Σ*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘(𝐹‘𝑖))) |
57 | 56 | mpteq2dva 5248 | . . 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 2778 | . 2 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ Σ*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))) |
59 | 8 | ralrimiva 3144 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑆) |
60 | dfiun2g 5035 | . . . . . 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 6968 | . . . . . . 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 4925 | . . . . 5 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (𝐹‘𝑛)}) |
65 | 61, 64 | eqtr4d 2778 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹) |
66 | 65 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑀‘∪ ran 𝐹)) |
67 | eqidd 2736 | . . . . 5 ⊢ (𝜑 → ℕ = ℕ) | |
68 | 67 | orcd 873 | . . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^(𝑖 + 1)))) |
69 | 37, 38, 68, 2, 8 | measiuns 34198 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = Σ*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)))) |
70 | 66, 69 | eqtr3d 2777 | . 2 ⊢ (𝜑 → (𝑀‘∪ ran 𝐹) = Σ*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)))) |
71 | 32, 58, 70 | 3brtr4d 5180 | 1 ⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝𝑡‘𝐽)(𝑀‘∪ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 ∖ cdif 3960 ⊆ wss 3963 ∪ cuni 4912 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 +∞cpnf 11290 ℕcn 12264 ℤcz 12611 [,]cicc 13387 ...cfz 13544 ..^cfzo 13691 ↾s cress 17274 TopOpenctopn 17468 ℝ*𝑠cxrs 17547 ⇝𝑡clm 23250 Σ*cesum 34008 sigAlgebracsiga 34089 measurescmeas 34176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-ordt 17548 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-scaf 20878 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-lm 23253 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tmd 24096 df-tgp 24097 df-tsms 24151 df-trg 24184 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 df-ii 24917 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-esum 34009 df-siga 34090 df-meas 34177 |
This theorem is referenced by: dstfrvclim1 34459 |
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