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Mirrors > Home > MPE Home > Th. List > metss2 | Structured version Visualization version GIF version |
Description: If the metric π· is "strongly finer" than πΆ (meaning that there is a positive real constant π such that πΆ(π₯, π¦) β€ π Β· π·(π₯, π¦)), then π· generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.) |
Ref | Expression |
---|---|
metequiv.3 | β’ π½ = (MetOpenβπΆ) |
metequiv.4 | β’ πΎ = (MetOpenβπ·) |
metss2.1 | β’ (π β πΆ β (Metβπ)) |
metss2.2 | β’ (π β π· β (Metβπ)) |
metss2.3 | β’ (π β π β β+) |
metss2.4 | β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯πΆπ¦) β€ (π Β· (π₯π·π¦))) |
Ref | Expression |
---|---|
metss2 | β’ (π β π½ β πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . . . 5 β’ ((π₯ β π β§ π β β+) β π β β+) | |
2 | metss2.3 | . . . . 5 β’ (π β π β β+) | |
3 | rpdivcl 13039 | . . . . 5 β’ ((π β β+ β§ π β β+) β (π / π ) β β+) | |
4 | 1, 2, 3 | syl2anr 595 | . . . 4 β’ ((π β§ (π₯ β π β§ π β β+)) β (π / π ) β β+) |
5 | metequiv.3 | . . . . 5 β’ π½ = (MetOpenβπΆ) | |
6 | metequiv.4 | . . . . 5 β’ πΎ = (MetOpenβπ·) | |
7 | metss2.1 | . . . . 5 β’ (π β πΆ β (Metβπ)) | |
8 | metss2.2 | . . . . 5 β’ (π β π· β (Metβπ)) | |
9 | metss2.4 | . . . . 5 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯πΆπ¦) β€ (π Β· (π₯π·π¦))) | |
10 | 5, 6, 7, 8, 2, 9 | metss2lem 24440 | . . . 4 β’ ((π β§ (π₯ β π β§ π β β+)) β (π₯(ballβπ·)(π / π )) β (π₯(ballβπΆ)π)) |
11 | oveq2 7434 | . . . . . 6 β’ (π = (π / π ) β (π₯(ballβπ·)π ) = (π₯(ballβπ·)(π / π ))) | |
12 | 11 | sseq1d 4013 | . . . . 5 β’ (π = (π / π ) β ((π₯(ballβπ·)π ) β (π₯(ballβπΆ)π) β (π₯(ballβπ·)(π / π )) β (π₯(ballβπΆ)π))) |
13 | 12 | rspcev 3611 | . . . 4 β’ (((π / π ) β β+ β§ (π₯(ballβπ·)(π / π )) β (π₯(ballβπΆ)π)) β βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π)) |
14 | 4, 10, 13 | syl2anc 582 | . . 3 β’ ((π β§ (π₯ β π β§ π β β+)) β βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π)) |
15 | 14 | ralrimivva 3198 | . 2 β’ (π β βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π)) |
16 | metxmet 24260 | . . . 4 β’ (πΆ β (Metβπ) β πΆ β (βMetβπ)) | |
17 | 7, 16 | syl 17 | . . 3 β’ (π β πΆ β (βMetβπ)) |
18 | metxmet 24260 | . . . 4 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
19 | 8, 18 | syl 17 | . . 3 β’ (π β π· β (βMetβπ)) |
20 | 5, 6 | metss 24437 | . . 3 β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β (π½ β πΎ β βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π))) |
21 | 17, 19, 20 | syl2anc 582 | . 2 β’ (π β (π½ β πΎ β βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π))) |
22 | 15, 21 | mpbird 256 | 1 β’ (π β π½ β πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 βwrex 3067 β wss 3949 class class class wbr 5152 βcfv 6553 (class class class)co 7426 Β· cmul 11151 β€ cle 11287 / cdiv 11909 β+crp 13014 βMetcxmet 21271 Metcmet 21272 ballcbl 21273 MetOpencmopn 21276 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-bases 22869 |
This theorem is referenced by: equivcmet 25265 |
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