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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnrebl | Structured version Visualization version GIF version |
Description: A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.) |
Ref | Expression |
---|---|
opnrebl | β’ (π΄ β (topGenβran (,)) β (π΄ β β β§ βπ₯ β π΄ βπ¦ β β+ ((π₯ β π¦)(,)(π₯ + π¦)) β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . . 4 β’ ((abs β β ) βΎ (β Γ β)) = ((abs β β ) βΎ (β Γ β)) | |
2 | 1 | rexmet 24725 | . . 3 β’ ((abs β β ) βΎ (β Γ β)) β (βMetββ) |
3 | eqid 2727 | . . . . 5 β’ (MetOpenβ((abs β β ) βΎ (β Γ β))) = (MetOpenβ((abs β β ) βΎ (β Γ β))) | |
4 | 1, 3 | tgioo 24730 | . . . 4 β’ (topGenβran (,)) = (MetOpenβ((abs β β ) βΎ (β Γ β))) |
5 | 4 | elmopn2 24369 | . . 3 β’ (((abs β β ) βΎ (β Γ β)) β (βMetββ) β (π΄ β (topGenβran (,)) β (π΄ β β β§ βπ₯ β π΄ βπ¦ β β+ (π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) β π΄))) |
6 | 2, 5 | ax-mp 5 | . 2 β’ (π΄ β (topGenβran (,)) β (π΄ β β β§ βπ₯ β π΄ βπ¦ β β+ (π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) β π΄)) |
7 | ssel2 3975 | . . . . 5 β’ ((π΄ β β β§ π₯ β π΄) β π₯ β β) | |
8 | rpre 13020 | . . . . . . . 8 β’ (π¦ β β+ β π¦ β β) | |
9 | 1 | bl2ioo 24726 | . . . . . . . 8 β’ ((π₯ β β β§ π¦ β β) β (π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) = ((π₯ β π¦)(,)(π₯ + π¦))) |
10 | 8, 9 | sylan2 591 | . . . . . . 7 β’ ((π₯ β β β§ π¦ β β+) β (π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) = ((π₯ β π¦)(,)(π₯ + π¦))) |
11 | 10 | sseq1d 4011 | . . . . . 6 β’ ((π₯ β β β§ π¦ β β+) β ((π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) β π΄ β ((π₯ β π¦)(,)(π₯ + π¦)) β π΄)) |
12 | 11 | rexbidva 3172 | . . . . 5 β’ (π₯ β β β (βπ¦ β β+ (π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) β π΄ β βπ¦ β β+ ((π₯ β π¦)(,)(π₯ + π¦)) β π΄)) |
13 | 7, 12 | syl 17 | . . . 4 β’ ((π΄ β β β§ π₯ β π΄) β (βπ¦ β β+ (π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) β π΄ β βπ¦ β β+ ((π₯ β π¦)(,)(π₯ + π¦)) β π΄)) |
14 | 13 | ralbidva 3171 | . . 3 β’ (π΄ β β β (βπ₯ β π΄ βπ¦ β β+ (π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) β π΄ β βπ₯ β π΄ βπ¦ β β+ ((π₯ β π¦)(,)(π₯ + π¦)) β π΄)) |
15 | 14 | pm5.32i 573 | . 2 β’ ((π΄ β β β§ βπ₯ β π΄ βπ¦ β β+ (π₯(ballβ((abs β β ) βΎ (β Γ β)))π¦) β π΄) β (π΄ β β β§ βπ₯ β π΄ βπ¦ β β+ ((π₯ β π¦)(,)(π₯ + π¦)) β π΄)) |
16 | 6, 15 | bitri 274 | 1 β’ (π΄ β (topGenβran (,)) β (π΄ β β β§ βπ₯ β π΄ βπ¦ β β+ ((π₯ β π¦)(,)(π₯ + π¦)) β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3057 βwrex 3066 β wss 3947 Γ cxp 5678 ran crn 5681 βΎ cres 5682 β ccom 5684 βcfv 6551 (class class class)co 7424 βcr 11143 + caddc 11147 β cmin 11480 β+crp 13012 (,)cioo 13362 abscabs 15219 topGenctg 17424 βMetcxmet 21269 ballcbl 21271 MetOpencmopn 21274 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-inf 9472 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ioo 13366 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-topgen 17430 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-bases 22867 |
This theorem is referenced by: (None) |
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