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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dya2icoseg2 | Structured version Visualization version GIF version |
Description: For any point and any open interval of ℝ containing that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-2017.) |
Ref | Expression |
---|---|
sxbrsiga.0 | ⊢ 𝐽 = (topGen‘ran (,)) |
dya2ioc.1 | ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) |
Ref | Expression |
---|---|
dya2icoseg2 | ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sxbrsiga.0 | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) | |
2 | dya2ioc.1 | . . . . . 6 ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛)))) | |
3 | eqid 2772 | . . . . . 6 ⊢ (⌊‘(1 − (2 logb 𝑑))) = (⌊‘(1 − (2 logb 𝑑))) | |
4 | 1, 2, 3 | dya2icoseg 31137 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ+) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)))) |
5 | 4 | ralrimiva 3126 | . . . 4 ⊢ (𝑋 ∈ ℝ → ∀𝑑 ∈ ℝ+ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)))) |
6 | 5 | 3ad2ant1 1113 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑑 ∈ ℝ+ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)))) |
7 | simp3 1118 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝑋 ∈ 𝐸) | |
8 | iooex 12570 | . . . . . . . . . 10 ⊢ (,) ∈ V | |
9 | 8 | rnex 7426 | . . . . . . . . 9 ⊢ ran (,) ∈ V |
10 | bastg 21268 | . . . . . . . . 9 ⊢ (ran (,) ∈ V → ran (,) ⊆ (topGen‘ran (,))) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . 8 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
12 | simp2 1117 | . . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ ran (,)) | |
13 | 11, 12 | sseldi 3852 | . . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ (topGen‘ran (,))) |
14 | 13, 1 | syl6eleqr 2871 | . . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → 𝐸 ∈ 𝐽) |
15 | eqid 2772 | . . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
16 | 15 | rexmet 23092 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) |
17 | recms 23676 | . . . . . . . . . . 11 ⊢ ℝfld ∈ CMetSp | |
18 | cmsms 23644 | . . . . . . . . . . 11 ⊢ (ℝfld ∈ CMetSp → ℝfld ∈ MetSp) | |
19 | msxms 22757 | . . . . . . . . . . 11 ⊢ (ℝfld ∈ MetSp → ℝfld ∈ ∞MetSp) | |
20 | 17, 18, 19 | mp2b 10 | . . . . . . . . . 10 ⊢ ℝfld ∈ ∞MetSp |
21 | retopn 23675 | . . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
22 | 1, 21 | eqtri 2796 | . . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℝfld) |
23 | rebase 20442 | . . . . . . . . . . 11 ⊢ ℝ = (Base‘ℝfld) | |
24 | reds 20452 | . . . . . . . . . . . 12 ⊢ (abs ∘ − ) = (dist‘ℝfld) | |
25 | 24 | reseq1i 5684 | . . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((dist‘ℝfld) ↾ (ℝ × ℝ)) |
26 | 22, 23, 25 | xmstopn 22754 | . . . . . . . . . 10 ⊢ (ℝfld ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))) |
27 | 20, 26 | ax-mp 5 | . . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) |
28 | 27 | elmopn2 22748 | . . . . . . . 8 ⊢ (((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) → (𝐸 ∈ 𝐽 ↔ (𝐸 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ+ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸))) |
29 | 16, 28 | ax-mp 5 | . . . . . . 7 ⊢ (𝐸 ∈ 𝐽 ↔ (𝐸 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ+ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)) |
30 | 29 | simprbi 489 | . . . . . 6 ⊢ (𝐸 ∈ 𝐽 → ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ+ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸) |
31 | 14, 30 | syl 17 | . . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ+ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸) |
32 | oveq1 6977 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑)) | |
33 | 32 | sseq1d 3884 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)) |
34 | 33 | rexbidv 3236 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑑 ∈ ℝ+ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ+ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸)) |
35 | 34 | rspcva 3527 | . . . . 5 ⊢ ((𝑋 ∈ 𝐸 ∧ ∀𝑥 ∈ 𝐸 ∃𝑑 ∈ ℝ+ (𝑥(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸) → ∃𝑑 ∈ ℝ+ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸) |
36 | 7, 31, 35 | syl2anc 576 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ+ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸) |
37 | rpre 12205 | . . . . . . 7 ⊢ (𝑑 ∈ ℝ+ → 𝑑 ∈ ℝ) | |
38 | 15 | bl2ioo 23093 | . . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) = ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) |
39 | 38 | sseq1d 3884 | . . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) |
40 | 37, 39 | sylan2 583 | . . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑑 ∈ ℝ+) → ((𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) |
41 | 40 | rexbidva 3235 | . . . . 5 ⊢ (𝑋 ∈ ℝ → (∃𝑑 ∈ ℝ+ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ+ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) |
42 | 41 | 3ad2ant1 1113 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → (∃𝑑 ∈ ℝ+ (𝑋(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑑) ⊆ 𝐸 ↔ ∃𝑑 ∈ ℝ+ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) |
43 | 36, 42 | mpbid 224 | . . 3 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ+ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) |
44 | r19.29 3194 | . . 3 ⊢ ((∀𝑑 ∈ ℝ+ ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ∃𝑑 ∈ ℝ+ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑑 ∈ ℝ+ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) | |
45 | 6, 43, 44 | syl2anc 576 | . 2 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑑 ∈ ℝ+ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) |
46 | r19.41v 3282 | . . . 4 ⊢ (∃𝑏 ∈ ran 𝐼((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) ↔ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) | |
47 | sstr 3862 | . . . . . . 7 ⊢ ((𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → 𝑏 ⊆ 𝐸) | |
48 | 47 | anim2i 607 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑏 ∧ (𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸)) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) |
49 | 48 | anassrs 460 | . . . . 5 ⊢ (((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → (𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) |
50 | 49 | reximi 3184 | . . . 4 ⊢ (∃𝑏 ∈ ran 𝐼((𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) |
51 | 46, 50 | sylbir 227 | . . 3 ⊢ ((∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) |
52 | 51 | rexlimivw 3221 | . 2 ⊢ (∃𝑑 ∈ ℝ+ (∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑))) ∧ ((𝑋 − 𝑑)(,)(𝑋 + 𝑑)) ⊆ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) |
53 | 45, 52 | syl 17 | 1 ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ∀wral 3082 ∃wrex 3083 Vcvv 3409 ⊆ wss 3825 × cxp 5398 ran crn 5401 ↾ cres 5402 ∘ ccom 5404 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 ℝcr 10326 1c1 10328 + caddc 10330 − cmin 10662 / cdiv 11090 2c2 11488 ℤcz 11786 ℝ+crp 12197 (,)cioo 12547 [,)cico 12549 ⌊cfl 12968 ↑cexp 13237 abscabs 14444 distcds 16420 TopOpenctopn 16541 topGenctg 16557 ∞Metcxmet 20222 ballcbl 20224 MetOpencmopn 20227 ℝfldcrefld 20440 ∞MetSpcxms 22620 MetSpcms 22621 CMetSpccms 23628 logb clogb 25033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-card 9154 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ioo 12551 df-ioc 12552 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 df-fac 13442 df-bc 13471 df-hash 13499 df-shft 14277 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-limsup 14679 df-clim 14696 df-rlim 14697 df-sum 14894 df-ef 15271 df-sin 15273 df-cos 15274 df-pi 15276 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-mulg 18002 df-cntz 18208 df-cmn 18658 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-fbas 20234 df-fg 20235 df-cnfld 20238 df-refld 20441 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cld 21321 df-ntr 21322 df-cls 21323 df-nei 21400 df-lp 21438 df-perf 21439 df-cn 21529 df-cnp 21530 df-haus 21617 df-cmp 21689 df-tx 21864 df-hmeo 22057 df-fil 22148 df-fm 22240 df-flim 22241 df-flf 22242 df-fcls 22243 df-xms 22623 df-ms 22624 df-tms 22625 df-cncf 23179 df-cfil 23551 df-cmet 23553 df-cms 23631 df-limc 24157 df-dv 24158 df-log 24831 df-cxp 24832 df-logb 25034 |
This theorem is referenced by: dya2iocnrect 31141 |
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