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Mirrors > Home > MPE Home > Th. List > blssec | Structured version Visualization version GIF version |
Description: A ball centered at 𝑃 is contained in the set of points finitely separated from 𝑃. This is just an application of ssbl 23275 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
xmeter.1 | ⊢ ∼ = (◡𝐷 “ ℝ) |
Ref | Expression |
---|---|
blssec | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ [𝑃] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfge 12687 | . . . . 5 ⊢ (𝑆 ∈ ℝ* → 𝑆 ≤ +∞) | |
2 | 1 | adantl 485 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑆 ∈ ℝ*) → 𝑆 ≤ +∞) |
3 | pnfxr 10852 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
4 | ssbl 23275 | . . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑆 ∈ ℝ* ∧ +∞ ∈ ℝ*) ∧ 𝑆 ≤ +∞) → (𝑃(ball‘𝐷)𝑆) ⊆ (𝑃(ball‘𝐷)+∞)) | |
5 | 4 | 3expia 1123 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑆 ∈ ℝ* ∧ +∞ ∈ ℝ*)) → (𝑆 ≤ +∞ → (𝑃(ball‘𝐷)𝑆) ⊆ (𝑃(ball‘𝐷)+∞))) |
6 | 3, 5 | mpanr2 704 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑆 ∈ ℝ*) → (𝑆 ≤ +∞ → (𝑃(ball‘𝐷)𝑆) ⊆ (𝑃(ball‘𝐷)+∞))) |
7 | 2, 6 | mpd 15 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ (𝑃(ball‘𝐷)+∞)) |
8 | 7 | 3impa 1112 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ (𝑃(ball‘𝐷)+∞)) |
9 | xmeter.1 | . . . 4 ⊢ ∼ = (◡𝐷 “ ℝ) | |
10 | 9 | xmetec 23286 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → [𝑃] ∼ = (𝑃(ball‘𝐷)+∞)) |
11 | 10 | 3adant3 1134 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → [𝑃] ∼ = (𝑃(ball‘𝐷)+∞)) |
12 | 8, 11 | sseqtrrd 3928 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ [𝑃] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 class class class wbr 5039 ◡ccnv 5535 “ cima 5539 ‘cfv 6358 (class class class)co 7191 [cec 8367 ℝcr 10693 +∞cpnf 10829 ℝ*cxr 10831 ≤ cle 10833 ∞Metcxmet 20302 ballcbl 20304 |
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 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-ec 8371 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-2 11858 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-psmet 20309 df-xmet 20310 df-bl 20312 |
This theorem is referenced by: xmetresbl 23289 xrsblre 23662 isbndx 35626 |
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