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Mirrors > Home > MPE Home > Th. List > blcls | Structured version Visualization version GIF version |
Description: The closure of an open ball in a metric space is contained in the corresponding closed ball. (Equality need not hold; for example, with the discrete metric, the closed ball of radius 1 is the whole space, but the open ball of radius 1 is just a point, whose closure is also a point.) (Contributed by Mario Carneiro, 31-Dec-2013.) |
Ref | Expression |
---|---|
mopni.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
blcld.3 | ⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} |
Ref | Expression |
---|---|
blcls | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mopni.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | blcld.3 | . . 3 ⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} | |
3 | 1, 2 | blcld 22120 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑆 ∈ (Clsd‘𝐽)) |
4 | blssm 22033 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) | |
5 | elbl 22003 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑧 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑧 ∈ 𝑋 ∧ (𝑃𝐷𝑧) < 𝑅))) | |
6 | xmetcl 21946 | . . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) ∈ ℝ*) | |
7 | 6 | 3expa 1257 | . . . . . . . . 9 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) ∈ ℝ*) |
8 | 7 | 3adantl3 1212 | . . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) ∈ ℝ*) |
9 | simpl3 1059 | . . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝑧 ∈ 𝑋) → 𝑅 ∈ ℝ*) | |
10 | xrltle 11858 | . . . . . . . 8 ⊢ (((𝑃𝐷𝑧) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → ((𝑃𝐷𝑧) < 𝑅 → (𝑃𝐷𝑧) ≤ 𝑅)) | |
11 | 8, 9, 10 | syl2anc 691 | . . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐷𝑧) < 𝑅 → (𝑃𝐷𝑧) ≤ 𝑅)) |
12 | 11 | expimpd 627 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ((𝑧 ∈ 𝑋 ∧ (𝑃𝐷𝑧) < 𝑅) → (𝑃𝐷𝑧) ≤ 𝑅)) |
13 | 5, 12 | sylbid 229 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑧 ∈ (𝑃(ball‘𝐷)𝑅) → (𝑃𝐷𝑧) ≤ 𝑅)) |
14 | 13 | ralrimiv 2948 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ∀𝑧 ∈ (𝑃(ball‘𝐷)𝑅)(𝑃𝐷𝑧) ≤ 𝑅) |
15 | ssrab 3643 | . . . 4 ⊢ ((𝑃(ball‘𝐷)𝑅) ⊆ {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ↔ ((𝑃(ball‘𝐷)𝑅) ⊆ 𝑋 ∧ ∀𝑧 ∈ (𝑃(ball‘𝐷)𝑅)(𝑃𝐷𝑧) ≤ 𝑅)) | |
16 | 4, 14, 15 | sylanbrc 695 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}) |
17 | 16, 2 | syl6sseqr 3615 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑆) |
18 | eqid 2610 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
19 | 18 | clsss2 20686 | . 2 ⊢ ((𝑆 ∈ (Clsd‘𝐽) ∧ (𝑃(ball‘𝐷)𝑅) ⊆ 𝑆) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ 𝑆) |
20 | 3, 17, 19 | syl2anc 691 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ((cls‘𝐽)‘(𝑃(ball‘𝐷)𝑅)) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 ∞Metcxmt 19552 ballcbl 19554 MetOpencmopn 19557 Clsdccld 20630 clsccl 20632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-cls 20635 |
This theorem is referenced by: blsscls 22122 cnllycmp 22563 cncmet 22927 |
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