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Mirrors > Home > MPE Home > Th. List > lebnumlem2 | Structured version Visualization version GIF version |
Description: Lemma for lebnum 24711. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24593, the function πΉ is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.) |
Ref | Expression |
---|---|
lebnum.j | β’ π½ = (MetOpenβπ·) |
lebnum.d | β’ (π β π· β (Metβπ)) |
lebnum.c | β’ (π β π½ β Comp) |
lebnum.s | β’ (π β π β π½) |
lebnum.u | β’ (π β π = βͺ π) |
lebnumlem1.u | β’ (π β π β Fin) |
lebnumlem1.n | β’ (π β Β¬ π β π) |
lebnumlem1.f | β’ πΉ = (π¦ β π β¦ Ξ£π β π inf(ran (π§ β (π β π) β¦ (π¦π·π§)), β*, < )) |
lebnumlem2.k | β’ πΎ = (topGenβran (,)) |
Ref | Expression |
---|---|
lebnumlem2 | β’ (π β πΉ β (π½ Cn πΎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lebnumlem1.f | . . . 4 β’ πΉ = (π¦ β π β¦ Ξ£π β π inf(ran (π§ β (π β π) β¦ (π¦π·π§)), β*, < )) | |
2 | eqid 2731 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
3 | lebnum.d | . . . . . . 7 β’ (π β π· β (Metβπ)) | |
4 | metxmet 24061 | . . . . . . 7 β’ (π· β (Metβπ) β π· β (βMetβπ)) | |
5 | 3, 4 | syl 17 | . . . . . 6 β’ (π β π· β (βMetβπ)) |
6 | lebnum.j | . . . . . . 7 β’ π½ = (MetOpenβπ·) | |
7 | 6 | mopntopon 24166 | . . . . . 6 β’ (π· β (βMetβπ) β π½ β (TopOnβπ)) |
8 | 5, 7 | syl 17 | . . . . 5 β’ (π β π½ β (TopOnβπ)) |
9 | lebnumlem1.u | . . . . 5 β’ (π β π β Fin) | |
10 | 3 | adantr 480 | . . . . . 6 β’ ((π β§ π β π) β π· β (Metβπ)) |
11 | difssd 4133 | . . . . . 6 β’ ((π β§ π β π) β (π β π) β π) | |
12 | 5 | adantr 480 | . . . . . . . . 9 β’ ((π β§ π β π) β π· β (βMetβπ)) |
13 | 12, 7 | syl 17 | . . . . . . . 8 β’ ((π β§ π β π) β π½ β (TopOnβπ)) |
14 | lebnum.s | . . . . . . . . 9 β’ (π β π β π½) | |
15 | 14 | sselda 3983 | . . . . . . . 8 β’ ((π β§ π β π) β π β π½) |
16 | toponss 22650 | . . . . . . . 8 β’ ((π½ β (TopOnβπ) β§ π β π½) β π β π) | |
17 | 13, 15, 16 | syl2anc 583 | . . . . . . 7 β’ ((π β§ π β π) β π β π) |
18 | lebnumlem1.n | . . . . . . . . . 10 β’ (π β Β¬ π β π) | |
19 | eleq1 2820 | . . . . . . . . . . 11 β’ (π = π β (π β π β π β π)) | |
20 | 19 | notbid 317 | . . . . . . . . . 10 β’ (π = π β (Β¬ π β π β Β¬ π β π)) |
21 | 18, 20 | syl5ibrcom 246 | . . . . . . . . 9 β’ (π β (π = π β Β¬ π β π)) |
22 | 21 | necon2ad 2954 | . . . . . . . 8 β’ (π β (π β π β π β π)) |
23 | 22 | imp 406 | . . . . . . 7 β’ ((π β§ π β π) β π β π) |
24 | pssdifn0 4366 | . . . . . . 7 β’ ((π β π β§ π β π) β (π β π) β β ) | |
25 | 17, 23, 24 | syl2anc 583 | . . . . . 6 β’ ((π β§ π β π) β (π β π) β β ) |
26 | eqid 2731 | . . . . . . 7 β’ (π¦ β π β¦ inf(ran (π§ β (π β π) β¦ (π¦π·π§)), β*, < )) = (π¦ β π β¦ inf(ran (π§ β (π β π) β¦ (π¦π·π§)), β*, < )) | |
27 | 26, 6, 2 | metdscn2 24594 | . . . . . 6 β’ ((π· β (Metβπ) β§ (π β π) β π β§ (π β π) β β ) β (π¦ β π β¦ inf(ran (π§ β (π β π) β¦ (π¦π·π§)), β*, < )) β (π½ Cn (TopOpenββfld))) |
28 | 10, 11, 25, 27 | syl3anc 1370 | . . . . 5 β’ ((π β§ π β π) β (π¦ β π β¦ inf(ran (π§ β (π β π) β¦ (π¦π·π§)), β*, < )) β (π½ Cn (TopOpenββfld))) |
29 | 2, 8, 9, 28 | fsumcn 24609 | . . . 4 β’ (π β (π¦ β π β¦ Ξ£π β π inf(ran (π§ β (π β π) β¦ (π¦π·π§)), β*, < )) β (π½ Cn (TopOpenββfld))) |
30 | 1, 29 | eqeltrid 2836 | . . 3 β’ (π β πΉ β (π½ Cn (TopOpenββfld))) |
31 | 2 | cnfldtopon 24520 | . . . . 5 β’ (TopOpenββfld) β (TopOnββ) |
32 | 31 | a1i 11 | . . . 4 β’ (π β (TopOpenββfld) β (TopOnββ)) |
33 | lebnum.c | . . . . . . 7 β’ (π β π½ β Comp) | |
34 | lebnum.u | . . . . . . 7 β’ (π β π = βͺ π) | |
35 | 6, 3, 33, 14, 34, 9, 18, 1 | lebnumlem1 24708 | . . . . . 6 β’ (π β πΉ:πβΆβ+) |
36 | 35 | frnd 6726 | . . . . 5 β’ (π β ran πΉ β β+) |
37 | rpssre 12986 | . . . . 5 β’ β+ β β | |
38 | 36, 37 | sstrdi 3995 | . . . 4 β’ (π β ran πΉ β β) |
39 | ax-resscn 11170 | . . . . 5 β’ β β β | |
40 | 39 | a1i 11 | . . . 4 β’ (π β β β β) |
41 | cnrest2 23011 | . . . 4 β’ (((TopOpenββfld) β (TopOnββ) β§ ran πΉ β β β§ β β β) β (πΉ β (π½ Cn (TopOpenββfld)) β πΉ β (π½ Cn ((TopOpenββfld) βΎt β)))) | |
42 | 32, 38, 40, 41 | syl3anc 1370 | . . 3 β’ (π β (πΉ β (π½ Cn (TopOpenββfld)) β πΉ β (π½ Cn ((TopOpenββfld) βΎt β)))) |
43 | 30, 42 | mpbid 231 | . 2 β’ (π β πΉ β (π½ Cn ((TopOpenββfld) βΎt β))) |
44 | lebnumlem2.k | . . . 4 β’ πΎ = (topGenβran (,)) | |
45 | 2 | tgioo2 24540 | . . . 4 β’ (topGenβran (,)) = ((TopOpenββfld) βΎt β) |
46 | 44, 45 | eqtri 2759 | . . 3 β’ πΎ = ((TopOpenββfld) βΎt β) |
47 | 46 | oveq2i 7423 | . 2 β’ (π½ Cn πΎ) = (π½ Cn ((TopOpenββfld) βΎt β)) |
48 | 43, 47 | eleqtrrdi 2843 | 1 β’ (π β πΉ β (π½ Cn πΎ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 β wne 2939 β cdif 3946 β wss 3949 β c0 4323 βͺ cuni 4909 β¦ cmpt 5232 ran crn 5678 βcfv 6544 (class class class)co 7412 Fincfn 8942 infcinf 9439 βcc 11111 βcr 11112 β*cxr 11252 < clt 11253 β+crp 12979 (,)cioo 13329 Ξ£csu 15637 βΎt crest 17371 TopOpenctopn 17372 topGenctg 17388 βMetcxmet 21130 Metcmet 21131 MetOpencmopn 21135 βfldccnfld 21145 TopOnctopon 22633 Cn ccn 22949 Compccmp 23111 |
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 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-ec 8708 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-cn 22952 df-cnp 22953 df-tx 23287 df-hmeo 23480 df-xms 24047 df-ms 24048 df-tms 24049 |
This theorem is referenced by: lebnumlem3 24710 |
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