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| Mirrors > Home > MPE Home > Th. List > blssec | Structured version Visualization version GIF version | ||
| Description: A ball centered at π is contained in the set of points finitely separated from π. This is just an application of ssbl 23928 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.) |
| Ref | Expression |
|---|---|
| xmeter.1 | β’ βΌ = (β‘π· β β) |
| Ref | Expression |
|---|---|
| blssec | β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π) β [π] βΌ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pnfge 13109 | . . . . 5 β’ (π β β* β π β€ +β) | |
| 2 | 1 | adantl 482 | . . . 4 β’ (((π· β (βMetβπ) β§ π β π) β§ π β β*) β π β€ +β) |
| 3 | pnfxr 11267 | . . . . 5 β’ +β β β* | |
| 4 | ssbl 23928 | . . . . . 6 β’ (((π· β (βMetβπ) β§ π β π) β§ (π β β* β§ +β β β*) β§ π β€ +β) β (π(ballβπ·)π) β (π(ballβπ·)+β)) | |
| 5 | 4 | 3expia 1121 | . . . . 5 β’ (((π· β (βMetβπ) β§ π β π) β§ (π β β* β§ +β β β*)) β (π β€ +β β (π(ballβπ·)π) β (π(ballβπ·)+β))) |
| 6 | 3, 5 | mpanr2 702 | . . . 4 β’ (((π· β (βMetβπ) β§ π β π) β§ π β β*) β (π β€ +β β (π(ballβπ·)π) β (π(ballβπ·)+β))) |
| 7 | 2, 6 | mpd 15 | . . 3 β’ (((π· β (βMetβπ) β§ π β π) β§ π β β*) β (π(ballβπ·)π) β (π(ballβπ·)+β)) |
| 8 | 7 | 3impa 1110 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π) β (π(ballβπ·)+β)) |
| 9 | xmeter.1 | . . . 4 β’ βΌ = (β‘π· β β) | |
| 10 | 9 | xmetec 23939 | . . 3 β’ ((π· β (βMetβπ) β§ π β π) β [π] βΌ = (π(ballβπ·)+β)) |
| 11 | 10 | 3adant3 1132 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β [π] βΌ = (π(ballβπ·)+β)) |
| 12 | 8, 11 | sseqtrrd 4023 | 1 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π) β [π] βΌ ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β wss 3948 class class class wbr 5148 β‘ccnv 5675 β cima 5679 βcfv 6543 (class class class)co 7408 [cec 8700 βcr 11108 +βcpnf 11244 β*cxr 11246 β€ cle 11248 βMetcxmet 20928 ballcbl 20930 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-ec 8704 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-2 12274 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-psmet 20935 df-xmet 20936 df-bl 20938 |
| This theorem is referenced by: xmetresbl 23942 xrsblre 24326 isbndx 36645 |
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