![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > blfval | Structured version Visualization version GIF version |
Description: The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
blfval | β’ (π· β (βMetβπ) β (ballβπ·) = (π₯ β π, π β β* β¦ {π¦ β π β£ (π₯π·π¦) < π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetpsmet 24205 | . 2 β’ (π· β (βMetβπ) β π· β (PsMetβπ)) | |
2 | blfvalps 24240 | . 2 β’ (π· β (PsMetβπ) β (ballβπ·) = (π₯ β π, π β β* β¦ {π¦ β π β£ (π₯π·π¦) < π})) | |
3 | 1, 2 | syl 17 | 1 β’ (π· β (βMetβπ) β (ballβπ·) = (π₯ β π, π β β* β¦ {π¦ β π β£ (π₯π·π¦) < π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3426 class class class wbr 5141 βcfv 6536 (class class class)co 7404 β cmpo 7406 β*cxr 11248 < clt 11249 PsMetcpsmet 21220 βMetcxmet 21221 ballcbl 21223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-xr 11253 df-psmet 21228 df-xmet 21229 df-bl 21231 |
This theorem is referenced by: blval 24243 blf 24264 |
Copyright terms: Public domain | W3C validator |