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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 24159. (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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆)) |
4 | measbase 31458 | . . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
5 | 2, 4 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ∪ ran sigAlgebra) |
7 | meascnbl.2 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶𝑆) | |
8 | 7 | ffvelrnda 6853 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑆) |
9 | simpll 765 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑) | |
10 | fzossnn 13089 | . . . . . . . . 9 ⊢ (1..^𝑛) ⊆ ℕ | |
11 | simpr 487 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ (1..^𝑛)) | |
12 | 10, 11 | sseldi 3967 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ ℕ) |
13 | 7 | ffvelrnda 6853 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑆) |
14 | 9, 12, 13 | syl2anc 586 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1..^𝑛)) → (𝐹‘𝑘) ∈ 𝑆) |
15 | 14 | ralrimiva 3184 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) |
16 | sigaclfu2 31382 | . . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) | |
17 | 6, 15, 16 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) ∈ 𝑆) |
18 | difelsiga 31394 | . . . . 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 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) ∈ 𝑆) |
20 | measvxrge0 31466 | . . . 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 586 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) ∈ (0[,]+∞)) |
22 | fveq2 6672 | . . . . 5 ⊢ (𝑛 = 𝑜 → (𝐹‘𝑛) = (𝐹‘𝑜)) | |
23 | oveq2 7166 | . . . . . 6 ⊢ (𝑛 = 𝑜 → (1..^𝑛) = (1..^𝑜)) | |
24 | 23 | iuneq1d 4948 | . . . . 5 ⊢ (𝑛 = 𝑜 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘)) |
25 | 22, 24 | difeq12d 4102 | . . . 4 ⊢ (𝑛 = 𝑜 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘))) |
26 | 25 | fveq2d 6676 | . . 3 ⊢ (𝑛 = 𝑜 → (𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘((𝐹‘𝑜) ∖ ∪ 𝑘 ∈ (1..^𝑜)(𝐹‘𝑘)))) |
27 | fveq2 6672 | . . . . 5 ⊢ (𝑛 = 𝑝 → (𝐹‘𝑛) = (𝐹‘𝑝)) | |
28 | oveq2 7166 | . . . . . 6 ⊢ (𝑛 = 𝑝 → (1..^𝑛) = (1..^𝑝)) | |
29 | 28 | iuneq1d 4948 | . . . . 5 ⊢ (𝑛 = 𝑝 → ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘) = ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘)) |
30 | 27, 29 | difeq12d 4102 | . . . 4 ⊢ (𝑛 = 𝑝 → ((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)) = ((𝐹‘𝑝) ∖ ∪ 𝑘 ∈ (1..^𝑝)(𝐹‘𝑘))) |
31 | 30 | fveq2d 6676 | . . 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 31348 | . 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 31464 | . . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞)) | |
34 | 2, 33 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
35 | fcompt 6897 | . . . 4 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝐹:ℕ⟶𝑆) → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖)))) | |
36 | 34, 7, 35 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ (𝑀‘(𝐹‘𝑖)))) |
37 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑛(𝐹‘𝑘) | |
38 | fveq2 6672 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) | |
39 | simpr 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ) | |
40 | 39 | nnzd 12089 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ) |
41 | fzval3 13109 | . . . . . . . 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 870 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((1...𝑖) = ℕ ∨ (1...𝑖) = (1..^(𝑖 + 1)))) |
44 | 2 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑀 ∈ (measures‘𝑆)) |
45 | simpll 765 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝜑) | |
46 | fzossnn 13089 | . . . . . . . 8 ⊢ (1..^(𝑖 + 1)) ⊆ ℕ | |
47 | 42 | eleq2d 2900 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑛 ∈ (1...𝑖) ↔ 𝑛 ∈ (1..^(𝑖 + 1)))) |
48 | 47 | biimpa 479 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝑛 ∈ (1..^(𝑖 + 1))) |
49 | 46, 48 | sseldi 3967 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → 𝑛 ∈ ℕ) |
50 | 45, 49, 8 | syl2anc 586 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑖)) → (𝐹‘𝑛) ∈ 𝑆) |
51 | 37, 38, 43, 44, 50 | measiuns 31478 | . . . . 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 6517 | . . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ℕ) |
53 | meascnbl.3 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1))) | |
54 | 52, 53 | iuninc 30314 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛) = (𝐹‘𝑖)) |
55 | 54 | fveq2d 6676 | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑀‘∪ 𝑛 ∈ (1...𝑖)(𝐹‘𝑛)) = (𝑀‘(𝐹‘𝑖))) |
56 | 51, 55 | eqtr3d 2860 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → Σ*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))) = (𝑀‘(𝐹‘𝑖))) |
57 | 56 | mpteq2dva 5163 | . . 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 2861 | . 2 ⊢ (𝜑 → (𝑀 ∘ 𝐹) = (𝑖 ∈ ℕ ↦ Σ*𝑛 ∈ (1...𝑖)(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘))))) |
59 | 8 | ralrimiva 3184 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑆) |
60 | dfiun2g 4957 | . . . . . 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 6727 | . . . . . . 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 2861 | . . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹) |
66 | 65 | fveq2d 6676 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑀‘∪ ran 𝐹)) |
67 | eqidd 2824 | . . . . 5 ⊢ (𝜑 → ℕ = ℕ) | |
68 | 67 | orcd 869 | . . . 4 ⊢ (𝜑 → (ℕ = ℕ ∨ ℕ = (1..^(𝑖 + 1)))) |
69 | 37, 38, 68, 2, 8 | measiuns 31478 | . . 3 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = Σ*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)))) |
70 | 66, 69 | eqtr3d 2860 | . 2 ⊢ (𝜑 → (𝑀‘∪ ran 𝐹) = Σ*𝑛 ∈ ℕ(𝑀‘((𝐹‘𝑛) ∖ ∪ 𝑘 ∈ (1..^𝑛)(𝐹‘𝑘)))) |
71 | 32, 58, 70 | 3brtr4d 5100 | 1 ⊢ (𝜑 → (𝑀 ∘ 𝐹)(⇝𝑡‘𝐽)(𝑀‘∪ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 ∀wral 3140 ∃wrex 3141 ∖ cdif 3935 ⊆ wss 3938 ∪ cuni 4840 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 ∘ ccom 5561 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 +∞cpnf 10674 ℕcn 11640 ℤcz 11984 [,]cicc 12744 ...cfz 12895 ..^cfzo 13036 ↾s cress 16486 TopOpenctopn 16697 ℝ*𝑠cxrs 16775 ⇝𝑡clm 21836 Σ*cesum 31288 sigAlgebracsiga 31369 measurescmeas 31456 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-disj 5034 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-ordt 16776 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-ps 17812 df-tsr 17813 df-plusf 17853 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-abv 19590 df-lmod 19638 df-scaf 19639 df-sra 19946 df-rgmod 19947 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-lm 21839 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tmd 22682 df-tgp 22683 df-tsms 22737 df-trg 22770 df-xms 22932 df-ms 22933 df-tms 22934 df-nm 23194 df-ngp 23195 df-nrg 23197 df-nlm 23198 df-ii 23487 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-esum 31289 df-siga 31370 df-meas 31457 |
This theorem is referenced by: dstfrvclim1 31737 |
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