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| Mirrors > Home > ILE Home > Th. List > blssec | 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 13797 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 9785 | . . . . 5 β’ (π β β* β π β€ +β) | |
| 2 | 1 | adantl 277 | . . . 4 β’ (((π· β (βMetβπ) β§ π β π) β§ π β β*) β π β€ +β) |
| 3 | pnfxr 8006 | . . . . 5 β’ +β β β* | |
| 4 | ssbl 13797 | . . . . . 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 1205 | . . . . 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 438 | . . . 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 13 | . . 3 β’ (((π· β (βMetβπ) β§ π β π) β§ π β β*) β (π(ballβπ·)π) β (π(ballβπ·)+β)) |
| 8 | 7 | 3impa 1194 | . 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 13808 | . . 3 β’ ((π· β (βMetβπ) β§ π β π) β [π] βΌ = (π(ballβπ·)+β)) |
| 11 | 10 | 3adant3 1017 | . 2 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β [π] βΌ = (π(ballβπ·)+β)) |
| 12 | 8, 11 | sseqtrrd 3194 | 1 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π) β [π] βΌ ) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β§ w3a 978 = wceq 1353 β wcel 2148 β wss 3129 class class class wbr 4002 β‘ccnv 4624 β cima 4628 βcfv 5215 (class class class)co 5872 [cec 6530 βcr 7807 +βcpnf 7985 β*cxr 7987 β€ cle 7989 βMetcxmet 13309 ballcbl 13311 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 |
| This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-po 4295 df-iso 4296 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-ec 6534 df-map 6647 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-2 8974 df-xneg 9768 df-xadd 9769 df-psmet 13316 df-xmet 13317 df-bl 13319 |
| This theorem is referenced by: xmetresbl 13811 |
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