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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumcncf | Structured version Visualization version GIF version |
Description: The finite sum of continuous complex function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fsumcncf.x | β’ (π β π β β) |
fsumcncf.a | β’ (π β π΄ β Fin) |
fsumcncf.cncf | β’ ((π β§ π β π΄) β (π₯ β π β¦ π΅) β (πβcnββ)) |
Ref | Expression |
---|---|
fsumcncf | β’ (π β (π₯ β π β¦ Ξ£π β π΄ π΅) β (πβcnββ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . 3 β’ (TopOpenββfld) = (TopOpenββfld) | |
2 | 1 | cnfldtopon 24620 | . . . . 5 β’ (TopOpenββfld) β (TopOnββ) |
3 | 2 | a1i 11 | . . . 4 β’ (π β (TopOpenββfld) β (TopOnββ)) |
4 | fsumcncf.x | . . . 4 β’ (π β π β β) | |
5 | resttopon 22986 | . . . 4 β’ (((TopOpenββfld) β (TopOnββ) β§ π β β) β ((TopOpenββfld) βΎt π) β (TopOnβπ)) | |
6 | 3, 4, 5 | syl2anc 583 | . . 3 β’ (π β ((TopOpenββfld) βΎt π) β (TopOnβπ)) |
7 | fsumcncf.a | . . 3 β’ (π β π΄ β Fin) | |
8 | fsumcncf.cncf | . . . 4 β’ ((π β§ π β π΄) β (π₯ β π β¦ π΅) β (πβcnββ)) | |
9 | ssidd 3997 | . . . . . 6 β’ (π β β β β) | |
10 | eqid 2724 | . . . . . . 7 β’ ((TopOpenββfld) βΎt π) = ((TopOpenββfld) βΎt π) | |
11 | 1 | cnfldtop 24621 | . . . . . . . . 9 β’ (TopOpenββfld) β Top |
12 | unicntop 24623 | . . . . . . . . . 10 β’ β = βͺ (TopOpenββfld) | |
13 | 12 | restid 17377 | . . . . . . . . 9 β’ ((TopOpenββfld) β Top β ((TopOpenββfld) βΎt β) = (TopOpenββfld)) |
14 | 11, 13 | ax-mp 5 | . . . . . . . 8 β’ ((TopOpenββfld) βΎt β) = (TopOpenββfld) |
15 | 14 | eqcomi 2733 | . . . . . . 7 β’ (TopOpenββfld) = ((TopOpenββfld) βΎt β) |
16 | 1, 10, 15 | cncfcn 24751 | . . . . . 6 β’ ((π β β β§ β β β) β (πβcnββ) = (((TopOpenββfld) βΎt π) Cn (TopOpenββfld))) |
17 | 4, 9, 16 | syl2anc 583 | . . . . 5 β’ (π β (πβcnββ) = (((TopOpenββfld) βΎt π) Cn (TopOpenββfld))) |
18 | 17 | adantr 480 | . . . 4 β’ ((π β§ π β π΄) β (πβcnββ) = (((TopOpenββfld) βΎt π) Cn (TopOpenββfld))) |
19 | 8, 18 | eleqtrd 2827 | . . 3 β’ ((π β§ π β π΄) β (π₯ β π β¦ π΅) β (((TopOpenββfld) βΎt π) Cn (TopOpenββfld))) |
20 | 1, 6, 7, 19 | fsumcnf 44160 | . 2 β’ (π β (π₯ β π β¦ Ξ£π β π΄ π΅) β (((TopOpenββfld) βΎt π) Cn (TopOpenββfld))) |
21 | 20, 17 | eleqtrrd 2828 | 1 β’ (π β (π₯ β π β¦ Ξ£π β π΄ π΅) β (πβcnββ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3940 β¦ cmpt 5221 βcfv 6533 (class class class)co 7401 Fincfn 8934 βcc 11103 Ξ£csu 15628 βΎt crest 17364 TopOpenctopn 17365 βfldccnfld 21227 Topctop 22716 TopOnctopon 22733 Cn ccn 23049 βcnβccncf 24717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cn 23052 df-cnp 23053 df-tx 23387 df-hmeo 23580 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 |
This theorem is referenced by: dirkeritg 45269 etransclem34 45435 etransclem43 45444 |
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