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Mirrors > Home > MPE Home > Th. List > blpnf | Structured version Visualization version GIF version |
Description: The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
blpnf | β’ ((π· β (Metβπ) β§ π β π) β (π(ballβπ·)+β) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metxmet 24162 | . . . 4 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
2 | xblpnf 24224 | . . . 4 β’ ((π· β (βMetβπ) β§ π β π) β (π₯ β (π(ballβπ·)+β) β (π₯ β π β§ (ππ·π₯) β β))) | |
3 | 1, 2 | sylan 579 | . . 3 β’ ((π· β (Metβπ) β§ π β π) β (π₯ β (π(ballβπ·)+β) β (π₯ β π β§ (ππ·π₯) β β))) |
4 | metcl 24160 | . . . . 5 β’ ((π· β (Metβπ) β§ π β π β§ π₯ β π) β (ππ·π₯) β β) | |
5 | 4 | 3expia 1118 | . . . 4 β’ ((π· β (Metβπ) β§ π β π) β (π₯ β π β (ππ·π₯) β β)) |
6 | 5 | pm4.71d 561 | . . 3 β’ ((π· β (Metβπ) β§ π β π) β (π₯ β π β (π₯ β π β§ (ππ·π₯) β β))) |
7 | 3, 6 | bitr4d 282 | . 2 β’ ((π· β (Metβπ) β§ π β π) β (π₯ β (π(ballβπ·)+β) β π₯ β π)) |
8 | 7 | eqrdv 2722 | 1 β’ ((π· β (Metβπ) β§ π β π) β (π(ballβπ·)+β) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6533 (class class class)co 7401 βcr 11105 +βcpnf 11242 βMetcxmet 21213 Metcmet 21214 ballcbl 21215 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-2 12272 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 |
This theorem is referenced by: blssioo 24633 sblpnf 43558 |
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