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Mirrors > Home > MPE Home > Th. List > metequiv | Structured version Visualization version GIF version |
Description: Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
metequiv.3 | β’ π½ = (MetOpenβπΆ) |
metequiv.4 | β’ πΎ = (MetOpenβπ·) |
Ref | Expression |
---|---|
metequiv | β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β (π½ = πΎ β βπ₯ β π (βπ β β+ βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π) β§ βπ β β+ βπ β β+ (π₯(ballβπΆ)π) β (π₯(ballβπ·)π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metequiv.3 | . . . 4 β’ π½ = (MetOpenβπΆ) | |
2 | metequiv.4 | . . . 4 β’ πΎ = (MetOpenβπ·) | |
3 | 1, 2 | metss 23770 | . . 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))) |
4 | 2, 1 | metss 23770 | . . . 4 β’ ((π· β (βMetβπ) β§ πΆ β (βMetβπ)) β (πΎ β π½ β βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπΆ)π) β (π₯(ballβπ·)π))) |
5 | 4 | ancoms 459 | . . 3 β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β (πΎ β π½ β βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπΆ)π) β (π₯(ballβπ·)π))) |
6 | 3, 5 | anbi12d 631 | . 2 β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β ((π½ β πΎ β§ πΎ β π½) β (βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π) β§ βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπΆ)π) β (π₯(ballβπ·)π)))) |
7 | eqss 3947 | . 2 β’ (π½ = πΎ β (π½ β πΎ β§ πΎ β π½)) | |
8 | r19.26 3110 | . 2 β’ (βπ₯ β π (βπ β β+ βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π) β§ βπ β β+ βπ β β+ (π₯(ballβπΆ)π) β (π₯(ballβπ·)π)) β (βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π) β§ βπ₯ β π βπ β β+ βπ β β+ (π₯(ballβπΆ)π) β (π₯(ballβπ·)π))) | |
9 | 6, 7, 8 | 3bitr4g 313 | 1 β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β (π½ = πΎ β βπ₯ β π (βπ β β+ βπ β β+ (π₯(ballβπ·)π ) β (π₯(ballβπΆ)π) β§ βπ β β+ βπ β β+ (π₯(ballβπΆ)π) β (π₯(ballβπ·)π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1540 β wcel 2105 βwral 3061 βwrex 3070 β wss 3898 βcfv 6479 (class class class)co 7337 β+crp 12831 βMetcxmet 20688 ballcbl 20690 MetOpencmopn 20693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-inf 9300 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-n0 12335 df-z 12421 df-uz 12684 df-q 12790 df-rp 12832 df-xneg 12949 df-xadd 12950 df-xmul 12951 df-topgen 17251 df-psmet 20695 df-xmet 20696 df-bl 20698 df-mopn 20699 df-bases 22202 |
This theorem is referenced by: metequiv2 23772 |
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