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Mirrors > Home > MPE Home > Th. List > metcn | Structured version Visualization version GIF version |
Description: Two ways to say a mapping from metric πΆ to metric π· is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" π¦ there is a positive "delta" π§ such that a distance less than delta in πΆ maps to a distance less than epsilon in π·. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
metcn.2 | β’ π½ = (MetOpenβπΆ) |
metcn.4 | β’ πΎ = (MetOpenβπ·) |
Ref | Expression |
---|---|
metcn | β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β (πΉ β (π½ Cn πΎ) β (πΉ:πβΆπ β§ βπ₯ β π βπ¦ β β+ βπ§ β β+ βπ€ β π ((π₯πΆπ€) < π§ β ((πΉβπ₯)π·(πΉβπ€)) < π¦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metcn.2 | . . . 4 β’ π½ = (MetOpenβπΆ) | |
2 | 1 | mopntopon 24344 | . . 3 β’ (πΆ β (βMetβπ) β π½ β (TopOnβπ)) |
3 | metcn.4 | . . . 4 β’ πΎ = (MetOpenβπ·) | |
4 | 3 | mopntopon 24344 | . . 3 β’ (π· β (βMetβπ) β πΎ β (TopOnβπ)) |
5 | cncnp 23183 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β (πΉ β (π½ Cn πΎ) β (πΉ:πβΆπ β§ βπ₯ β π πΉ β ((π½ CnP πΎ)βπ₯)))) | |
6 | 2, 4, 5 | syl2an 595 | . 2 β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β (πΉ β (π½ Cn πΎ) β (πΉ:πβΆπ β§ βπ₯ β π πΉ β ((π½ CnP πΎ)βπ₯)))) |
7 | simplr 768 | . . . . 5 β’ ((((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β§ πΉ:πβΆπ) β§ π₯ β π) β πΉ:πβΆπ) | |
8 | 1, 3 | metcnp 24449 | . . . . . 6 β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ) β§ π₯ β π) β (πΉ β ((π½ CnP πΎ)βπ₯) β (πΉ:πβΆπ β§ βπ¦ β β+ βπ§ β β+ βπ€ β π ((π₯πΆπ€) < π§ β ((πΉβπ₯)π·(πΉβπ€)) < π¦)))) |
9 | 8 | ad4ant124 1171 | . . . . 5 β’ ((((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β§ πΉ:πβΆπ) β§ π₯ β π) β (πΉ β ((π½ CnP πΎ)βπ₯) β (πΉ:πβΆπ β§ βπ¦ β β+ βπ§ β β+ βπ€ β π ((π₯πΆπ€) < π§ β ((πΉβπ₯)π·(πΉβπ€)) < π¦)))) |
10 | 7, 9 | mpbirand 706 | . . . 4 β’ ((((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β§ πΉ:πβΆπ) β§ π₯ β π) β (πΉ β ((π½ CnP πΎ)βπ₯) β βπ¦ β β+ βπ§ β β+ βπ€ β π ((π₯πΆπ€) < π§ β ((πΉβπ₯)π·(πΉβπ€)) < π¦))) |
11 | 10 | ralbidva 3172 | . . 3 β’ (((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β§ πΉ:πβΆπ) β (βπ₯ β π πΉ β ((π½ CnP πΎ)βπ₯) β βπ₯ β π βπ¦ β β+ βπ§ β β+ βπ€ β π ((π₯πΆπ€) < π§ β ((πΉβπ₯)π·(πΉβπ€)) < π¦))) |
12 | 11 | pm5.32da 578 | . 2 β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β ((πΉ:πβΆπ β§ βπ₯ β π πΉ β ((π½ CnP πΎ)βπ₯)) β (πΉ:πβΆπ β§ βπ₯ β π βπ¦ β β+ βπ§ β β+ βπ€ β π ((π₯πΆπ€) < π§ β ((πΉβπ₯)π·(πΉβπ€)) < π¦)))) |
13 | 6, 12 | bitrd 279 | 1 β’ ((πΆ β (βMetβπ) β§ π· β (βMetβπ)) β (πΉ β (π½ Cn πΎ) β (πΉ:πβΆπ β§ βπ₯ β π βπ¦ β β+ βπ§ β β+ βπ€ β π ((π₯πΆπ€) < π§ β ((πΉβπ₯)π·(πΉβπ€)) < π¦)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3058 βwrex 3067 class class class wbr 5148 βΆwf 6544 βcfv 6548 (class class class)co 7420 < clt 11278 β+crp 13006 βMetcxmet 21263 MetOpencmopn 21268 TopOnctopon 22811 Cn ccn 23127 CnP ccnp 23128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-topgen 17424 df-psmet 21270 df-xmet 21271 df-bl 21273 df-mopn 21274 df-top 22795 df-topon 22812 df-bases 22848 df-cn 23130 df-cnp 23131 |
This theorem is referenced by: nrginvrcn 24608 nghmcn 24661 metdscn 24771 divcnOLD 24783 divcn 24785 cncfmet 24828 nmcvcn 30504 blocni 30614 hhcno 31713 hhcnf 31714 fmcncfil 33532 heicant 37128 |
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