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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > blssp | Structured version Visualization version GIF version |
Description: A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.) |
Ref | Expression |
---|---|
blssp.2 | β’ π = (π βΎ (π Γ π)) |
Ref | Expression |
---|---|
blssp | β’ (((π β (Metβπ) β§ π β π) β§ (π β π β§ π β β+)) β (π(ballβπ)π ) = ((π(ballβπ)π ) β© π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metxmet 23703 | . . 3 β’ (π β (Metβπ) β π β (βMetβπ)) | |
2 | 1 | ad2antrr 725 | . 2 β’ (((π β (Metβπ) β§ π β π) β§ (π β π β§ π β β+)) β π β (βMetβπ)) |
3 | simprl 770 | . . 3 β’ (((π β (Metβπ) β§ π β π) β§ (π β π β§ π β β+)) β π β π) | |
4 | simplr 768 | . . . 4 β’ (((π β (Metβπ) β§ π β π) β§ (π β π β§ π β β+)) β π β π) | |
5 | sseqin2 4180 | . . . 4 β’ (π β π β (π β© π) = π) | |
6 | 4, 5 | sylib 217 | . . 3 β’ (((π β (Metβπ) β§ π β π) β§ (π β π β§ π β β+)) β (π β© π) = π) |
7 | 3, 6 | eleqtrrd 2841 | . 2 β’ (((π β (Metβπ) β§ π β π) β§ (π β π β§ π β β+)) β π β (π β© π)) |
8 | rpxr 12931 | . . 3 β’ (π β β+ β π β β*) | |
9 | 8 | ad2antll 728 | . 2 β’ (((π β (Metβπ) β§ π β π) β§ (π β π β§ π β β+)) β π β β*) |
10 | blssp.2 | . . 3 β’ π = (π βΎ (π Γ π)) | |
11 | 10 | blres 23800 | . 2 β’ ((π β (βMetβπ) β§ π β (π β© π) β§ π β β*) β (π(ballβπ)π ) = ((π(ballβπ)π ) β© π)) |
12 | 2, 7, 9, 11 | syl3anc 1372 | 1 β’ (((π β (Metβπ) β§ π β π) β§ (π β π β§ π β β+)) β (π(ballβπ)π ) = ((π(ballβπ)π ) β© π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β© cin 3914 β wss 3915 Γ cxp 5636 βΎ cres 5640 βcfv 6501 (class class class)co 7362 β*cxr 11195 β+crp 12922 βMetcxmet 20797 Metcmet 20798 ballcbl 20799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-mulcl 11120 ax-i2m1 11126 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-rp 12923 df-xadd 13041 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 |
This theorem is referenced by: bndss 36274 |
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