![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > equivcmet | Structured version Visualization version GIF version |
Description: If two metrics are strongly equivalent, one is complete iff the other is. Unlike equivcau 24667, metss2 23871, this theorem does not have a one-directional form - it is possible for a metric πΆ that is strongly finer than the complete metric π· to be incomplete and vice versa. Consider π· = the metric on β induced by the usual homeomorphism from (0, 1) against the usual metric πΆ on β and against the discrete metric πΈ on β. Then both πΆ and πΈ are complete but π· is not, and πΆ is strongly finer than π·, which is strongly finer than πΈ. (Contributed by Mario Carneiro, 15-Sep-2015.) |
Ref | Expression |
---|---|
equivcmet.1 | β’ (π β πΆ β (Metβπ)) |
equivcmet.2 | β’ (π β π· β (Metβπ)) |
equivcmet.3 | β’ (π β π β β+) |
equivcmet.4 | β’ (π β π β β+) |
equivcmet.5 | β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯πΆπ¦) β€ (π Β· (π₯π·π¦))) |
equivcmet.6 | β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯π·π¦) β€ (π Β· (π₯πΆπ¦))) |
Ref | Expression |
---|---|
equivcmet | β’ (π β (πΆ β (CMetβπ) β π· β (CMetβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equivcmet.1 | . . . 4 β’ (π β πΆ β (Metβπ)) | |
2 | equivcmet.2 | . . . 4 β’ (π β π· β (Metβπ)) | |
3 | 1, 2 | 2thd 265 | . . 3 β’ (π β (πΆ β (Metβπ) β π· β (Metβπ))) |
4 | equivcmet.4 | . . . . . 6 β’ (π β π β β+) | |
5 | equivcmet.6 | . . . . . 6 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯π·π¦) β€ (π Β· (π₯πΆπ¦))) | |
6 | 2, 1, 4, 5 | equivcfil 24666 | . . . . 5 β’ (π β (CauFilβπΆ) β (CauFilβπ·)) |
7 | equivcmet.3 | . . . . . 6 β’ (π β π β β+) | |
8 | equivcmet.5 | . . . . . 6 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯πΆπ¦) β€ (π Β· (π₯π·π¦))) | |
9 | 1, 2, 7, 8 | equivcfil 24666 | . . . . 5 β’ (π β (CauFilβπ·) β (CauFilβπΆ)) |
10 | 6, 9 | eqssd 3962 | . . . 4 β’ (π β (CauFilβπΆ) = (CauFilβπ·)) |
11 | eqid 2737 | . . . . . . . 8 β’ (MetOpenβπΆ) = (MetOpenβπΆ) | |
12 | eqid 2737 | . . . . . . . 8 β’ (MetOpenβπ·) = (MetOpenβπ·) | |
13 | 11, 12, 1, 2, 7, 8 | metss2 23871 | . . . . . . 7 β’ (π β (MetOpenβπΆ) β (MetOpenβπ·)) |
14 | 12, 11, 2, 1, 4, 5 | metss2 23871 | . . . . . . 7 β’ (π β (MetOpenβπ·) β (MetOpenβπΆ)) |
15 | 13, 14 | eqssd 3962 | . . . . . 6 β’ (π β (MetOpenβπΆ) = (MetOpenβπ·)) |
16 | 15 | oveq1d 7373 | . . . . 5 β’ (π β ((MetOpenβπΆ) fLim π) = ((MetOpenβπ·) fLim π)) |
17 | 16 | neeq1d 3004 | . . . 4 β’ (π β (((MetOpenβπΆ) fLim π) β β β ((MetOpenβπ·) fLim π) β β )) |
18 | 10, 17 | raleqbidv 3320 | . . 3 β’ (π β (βπ β (CauFilβπΆ)((MetOpenβπΆ) fLim π) β β β βπ β (CauFilβπ·)((MetOpenβπ·) fLim π) β β )) |
19 | 3, 18 | anbi12d 632 | . 2 β’ (π β ((πΆ β (Metβπ) β§ βπ β (CauFilβπΆ)((MetOpenβπΆ) fLim π) β β ) β (π· β (Metβπ) β§ βπ β (CauFilβπ·)((MetOpenβπ·) fLim π) β β ))) |
20 | 11 | iscmet 24651 | . 2 β’ (πΆ β (CMetβπ) β (πΆ β (Metβπ) β§ βπ β (CauFilβπΆ)((MetOpenβπΆ) fLim π) β β )) |
21 | 12 | iscmet 24651 | . 2 β’ (π· β (CMetβπ) β (π· β (Metβπ) β§ βπ β (CauFilβπ·)((MetOpenβπ·) fLim π) β β )) |
22 | 19, 20, 21 | 3bitr4g 314 | 1 β’ (π β (πΆ β (CMetβπ) β π· β (CMetβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β wcel 2107 β wne 2944 βwral 3065 β c0 4283 class class class wbr 5106 βcfv 6497 (class class class)co 7358 Β· cmul 11057 β€ cle 11191 β+crp 12916 Metcmet 20785 MetOpencmopn 20789 fLim cflim 23288 CauFilccfil 24619 CMetccmet 24621 |
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-ico 13271 df-topgen 17326 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-fbas 20796 df-bases 22299 df-fil 23200 df-cfil 24622 df-cmet 24624 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |