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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 486 | . . . . 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 12941 | . . . . 5 β’ ((π β β+ β§ π β β+) β (π / π ) β β+) | |
4 | 1, 2, 3 | syl2anr 598 | . . . 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 23870 | . . . 4 β’ ((π β§ (π₯ β π β§ π β β+)) β (π₯(ballβπ·)(π / π )) β (π₯(ballβπΆ)π)) |
11 | oveq2 7366 | . . . . . 6 β’ (π = (π / π ) β (π₯(ballβπ·)π ) = (π₯(ballβπ·)(π / π ))) | |
12 | 11 | sseq1d 3976 | . . . . 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 3582 | . . . 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 585 | . . 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 23690 | . . . 4 β’ (πΆ β (Metβπ) β πΆ β (βMetβπ)) | |
17 | 7, 16 | syl 17 | . . 3 β’ (π β πΆ β (βMetβπ)) |
18 | metxmet 23690 | . . . 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 23867 | . . 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 585 | . 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 257 | 1 β’ (π β π½ β πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 βwrex 3074 β wss 3911 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Β· cmul 11057 β€ cle 11191 / cdiv 11813 β+crp 12916 βMetcxmet 20784 Metcmet 20785 ballcbl 20786 MetOpencmopn 20789 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-n0 12415 df-z 12501 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-topgen 17326 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-bases 22299 |
This theorem is referenced by: equivcmet 24684 |
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