![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpfinval | Structured version Visualization version GIF version |
Description: The value of the extended sum of a finite set of nonnegative finite terms. (Contributed by Thierry Arnoux, 28-Jun-2017.) (Proof shortened by AV, 25-Jul-2019.) |
Ref | Expression |
---|---|
esumpfinval.a | β’ (π β π΄ β Fin) |
esumpfinval.b | β’ ((π β§ π β π΄) β π΅ β (0[,)+β)) |
Ref | Expression |
---|---|
esumpfinval | β’ (π β Ξ£*π β π΄π΅ = Ξ£π β π΄ π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-esum 32667 | . . . 4 β’ Ξ£*π β π΄π΅ = βͺ ((β*π βΎs (0[,]+β)) tsums (π β π΄ β¦ π΅)) | |
2 | xrge0base 31918 | . . . . . 6 β’ (0[,]+β) = (Baseβ(β*π βΎs (0[,]+β))) | |
3 | xrge00 31919 | . . . . . 6 β’ 0 = (0gβ(β*π βΎs (0[,]+β))) | |
4 | xrge0cmn 20855 | . . . . . . 7 β’ (β*π βΎs (0[,]+β)) β CMnd | |
5 | 4 | a1i 11 | . . . . . 6 β’ (π β (β*π βΎs (0[,]+β)) β CMnd) |
6 | xrge0tps 32563 | . . . . . . 7 β’ (β*π βΎs (0[,]+β)) β TopSp | |
7 | 6 | a1i 11 | . . . . . 6 β’ (π β (β*π βΎs (0[,]+β)) β TopSp) |
8 | esumpfinval.a | . . . . . 6 β’ (π β π΄ β Fin) | |
9 | icossicc 13360 | . . . . . . . 8 β’ (0[,)+β) β (0[,]+β) | |
10 | esumpfinval.b | . . . . . . . 8 β’ ((π β§ π β π΄) β π΅ β (0[,)+β)) | |
11 | 9, 10 | sselid 3947 | . . . . . . 7 β’ ((π β§ π β π΄) β π΅ β (0[,]+β)) |
12 | 11 | fmpttd 7068 | . . . . . 6 β’ (π β (π β π΄ β¦ π΅):π΄βΆ(0[,]+β)) |
13 | eqid 2737 | . . . . . . 7 β’ (π β π΄ β¦ π΅) = (π β π΄ β¦ π΅) | |
14 | c0ex 11156 | . . . . . . . 8 β’ 0 β V | |
15 | 14 | a1i 11 | . . . . . . 7 β’ (π β 0 β V) |
16 | 13, 8, 10, 15 | fsuppmptdm 9323 | . . . . . 6 β’ (π β (π β π΄ β¦ π΅) finSupp 0) |
17 | xrge0topn 32564 | . . . . . . 7 β’ (TopOpenβ(β*π βΎs (0[,]+β))) = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
18 | 17 | eqcomi 2746 | . . . . . 6 β’ ((ordTopβ β€ ) βΎt (0[,]+β)) = (TopOpenβ(β*π βΎs (0[,]+β))) |
19 | xrhaus 22752 | . . . . . . . 8 β’ (ordTopβ β€ ) β Haus | |
20 | ovex 7395 | . . . . . . . 8 β’ (0[,]+β) β V | |
21 | resthaus 22735 | . . . . . . . 8 β’ (((ordTopβ β€ ) β Haus β§ (0[,]+β) β V) β ((ordTopβ β€ ) βΎt (0[,]+β)) β Haus) | |
22 | 19, 20, 21 | mp2an 691 | . . . . . . 7 β’ ((ordTopβ β€ ) βΎt (0[,]+β)) β Haus |
23 | 22 | a1i 11 | . . . . . 6 β’ (π β ((ordTopβ β€ ) βΎt (0[,]+β)) β Haus) |
24 | 2, 3, 5, 7, 8, 12, 16, 18, 23 | haustsmsid 23508 | . . . . 5 β’ (π β ((β*π βΎs (0[,]+β)) tsums (π β π΄ β¦ π΅)) = {((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅))}) |
25 | 24 | unieqd 4884 | . . . 4 β’ (π β βͺ ((β*π βΎs (0[,]+β)) tsums (π β π΄ β¦ π΅)) = βͺ {((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅))}) |
26 | 1, 25 | eqtrid 2789 | . . 3 β’ (π β Ξ£*π β π΄π΅ = βͺ {((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅))}) |
27 | ovex 7395 | . . . 4 β’ ((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅)) β V | |
28 | 27 | unisn 4892 | . . 3 β’ βͺ {((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅))} = ((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅)) |
29 | 26, 28 | eqtrdi 2793 | . 2 β’ (π β Ξ£*π β π΄π΅ = ((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅))) |
30 | 10 | fmpttd 7068 | . . 3 β’ (π β (π β π΄ β¦ π΅):π΄βΆ(0[,)+β)) |
31 | esumpfinvallem 32713 | . . 3 β’ ((π΄ β Fin β§ (π β π΄ β¦ π΅):π΄βΆ(0[,)+β)) β (βfld Ξ£g (π β π΄ β¦ π΅)) = ((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅))) | |
32 | 8, 30, 31 | syl2anc 585 | . 2 β’ (π β (βfld Ξ£g (π β π΄ β¦ π΅)) = ((β*π βΎs (0[,]+β)) Ξ£g (π β π΄ β¦ π΅))) |
33 | rge0ssre 13380 | . . . . 5 β’ (0[,)+β) β β | |
34 | ax-resscn 11115 | . . . . 5 β’ β β β | |
35 | 33, 34 | sstri 3958 | . . . 4 β’ (0[,)+β) β β |
36 | 35, 10 | sselid 3947 | . . 3 β’ ((π β§ π β π΄) β π΅ β β) |
37 | 8, 36 | gsumfsum 20880 | . 2 β’ (π β (βfld Ξ£g (π β π΄ β¦ π΅)) = Ξ£π β π΄ π΅) |
38 | 29, 32, 37 | 3eqtr2d 2783 | 1 β’ (π β Ξ£*π β π΄π΅ = Ξ£π β π΄ π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 {csn 4591 βͺ cuni 4870 β¦ cmpt 5193 βΆwf 6497 βcfv 6501 (class class class)co 7362 Fincfn 8890 βcc 11056 βcr 11057 0cc0 11058 +βcpnf 11193 β€ cle 11197 [,)cico 13273 [,]cicc 13274 Ξ£csu 15577 βΎs cress 17119 βΎt crest 17309 TopOpenctopn 17310 Ξ£g cgsu 17329 ordTopcordt 17388 β*π cxrs 17389 CMndccmn 19569 βfldccnfld 20812 TopSpctps 22297 Hauscha 22675 tsums ctsu 23493 Ξ£*cesum 32666 |
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-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
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 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-xadd 13041 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-ordt 17390 df-xrs 17391 df-ps 18462 df-tsr 18463 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-grp 18758 df-minusg 18759 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-cn 22594 df-haus 22682 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tsms 23494 df-esum 32667 |
This theorem is referenced by: hasheuni 32724 esumcvg 32725 sibfof 32980 |
Copyright terms: Public domain | W3C validator |