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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcocn | Structured version Visualization version GIF version |
Description: Lemma for esummulc2 32721 and co. Composing with a continuous function preserves extended sums. (Contributed by Thierry Arnoux, 29-Jun-2017.) |
Ref | Expression |
---|---|
esumcocn.j | β’ π½ = ((ordTopβ β€ ) βΎt (0[,]+β)) |
esumcocn.a | β’ (π β π΄ β π) |
esumcocn.b | β’ ((π β§ π β π΄) β π΅ β (0[,]+β)) |
esumcocn.1 | β’ (π β πΆ β (π½ Cn π½)) |
esumcocn.0 | β’ (π β (πΆβ0) = 0) |
esumcocn.f | β’ ((π β§ π₯ β (0[,]+β) β§ π¦ β (0[,]+β)) β (πΆβ(π₯ +π π¦)) = ((πΆβπ₯) +π (πΆβπ¦))) |
Ref | Expression |
---|---|
esumcocn | β’ (π β (πΆβΞ£*π β π΄π΅) = Ξ£*π β π΄(πΆβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1918 | . . 3 β’ β²ππ | |
2 | nfcv 2908 | . . 3 β’ β²ππ΄ | |
3 | esumcocn.a | . . 3 β’ (π β π΄ β π) | |
4 | esumcocn.1 | . . . . . 6 β’ (π β πΆ β (π½ Cn π½)) | |
5 | xrge0tps 32563 | . . . . . . . 8 β’ (β*π βΎs (0[,]+β)) β TopSp | |
6 | xrge0base 31918 | . . . . . . . . 9 β’ (0[,]+β) = (Baseβ(β*π βΎs (0[,]+β))) | |
7 | esumcocn.j | . . . . . . . . . 10 β’ π½ = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
8 | xrge0topn 32564 | . . . . . . . . . 10 β’ (TopOpenβ(β*π βΎs (0[,]+β))) = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
9 | 7, 8 | eqtr4i 2768 | . . . . . . . . 9 β’ π½ = (TopOpenβ(β*π βΎs (0[,]+β))) |
10 | 6, 9 | tpsuni 22301 | . . . . . . . 8 β’ ((β*π βΎs (0[,]+β)) β TopSp β (0[,]+β) = βͺ π½) |
11 | 5, 10 | ax-mp 5 | . . . . . . 7 β’ (0[,]+β) = βͺ π½ |
12 | 11, 11 | cnf 22613 | . . . . . 6 β’ (πΆ β (π½ Cn π½) β πΆ:(0[,]+β)βΆ(0[,]+β)) |
13 | 4, 12 | syl 17 | . . . . 5 β’ (π β πΆ:(0[,]+β)βΆ(0[,]+β)) |
14 | 13 | adantr 482 | . . . 4 β’ ((π β§ π β π΄) β πΆ:(0[,]+β)βΆ(0[,]+β)) |
15 | esumcocn.b | . . . 4 β’ ((π β§ π β π΄) β π΅ β (0[,]+β)) | |
16 | 14, 15 | ffvelcdmd 7041 | . . 3 β’ ((π β§ π β π΄) β (πΆβπ΅) β (0[,]+β)) |
17 | xrge0cmn 20855 | . . . . . 6 β’ (β*π βΎs (0[,]+β)) β CMnd | |
18 | 17 | a1i 11 | . . . . 5 β’ (π β (β*π βΎs (0[,]+β)) β CMnd) |
19 | 5 | a1i 11 | . . . . 5 β’ (π β (β*π βΎs (0[,]+β)) β TopSp) |
20 | cmnmnd 19586 | . . . . . . . 8 β’ ((β*π βΎs (0[,]+β)) β CMnd β (β*π βΎs (0[,]+β)) β Mnd) | |
21 | 17, 20 | ax-mp 5 | . . . . . . 7 β’ (β*π βΎs (0[,]+β)) β Mnd |
22 | 21 | a1i 11 | . . . . . 6 β’ (π β (β*π βΎs (0[,]+β)) β Mnd) |
23 | esumcocn.f | . . . . . . . 8 β’ ((π β§ π₯ β (0[,]+β) β§ π¦ β (0[,]+β)) β (πΆβ(π₯ +π π¦)) = ((πΆβπ₯) +π (πΆβπ¦))) | |
24 | 23 | 3expib 1123 | . . . . . . 7 β’ (π β ((π₯ β (0[,]+β) β§ π¦ β (0[,]+β)) β (πΆβ(π₯ +π π¦)) = ((πΆβπ₯) +π (πΆβπ¦)))) |
25 | 24 | ralrimivv 3196 | . . . . . 6 β’ (π β βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)(πΆβ(π₯ +π π¦)) = ((πΆβπ₯) +π (πΆβπ¦))) |
26 | esumcocn.0 | . . . . . 6 β’ (π β (πΆβ0) = 0) | |
27 | xrge0plusg 31920 | . . . . . . . 8 β’ +π = (+gβ(β*π βΎs (0[,]+β))) | |
28 | xrge00 31919 | . . . . . . . 8 β’ 0 = (0gβ(β*π βΎs (0[,]+β))) | |
29 | 6, 6, 27, 27, 28, 28 | ismhm 18610 | . . . . . . 7 β’ (πΆ β ((β*π βΎs (0[,]+β)) MndHom (β*π βΎs (0[,]+β))) β (((β*π βΎs (0[,]+β)) β Mnd β§ (β*π βΎs (0[,]+β)) β Mnd) β§ (πΆ:(0[,]+β)βΆ(0[,]+β) β§ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)(πΆβ(π₯ +π π¦)) = ((πΆβπ₯) +π (πΆβπ¦)) β§ (πΆβ0) = 0))) |
30 | 29 | biimpri 227 | . . . . . 6 β’ ((((β*π βΎs (0[,]+β)) β Mnd β§ (β*π βΎs (0[,]+β)) β Mnd) β§ (πΆ:(0[,]+β)βΆ(0[,]+β) β§ βπ₯ β (0[,]+β)βπ¦ β (0[,]+β)(πΆβ(π₯ +π π¦)) = ((πΆβπ₯) +π (πΆβπ¦)) β§ (πΆβ0) = 0)) β πΆ β ((β*π βΎs (0[,]+β)) MndHom (β*π βΎs (0[,]+β)))) |
31 | 22, 22, 13, 25, 26, 30 | syl23anc 1378 | . . . . 5 β’ (π β πΆ β ((β*π βΎs (0[,]+β)) MndHom (β*π βΎs (0[,]+β)))) |
32 | eqidd 2738 | . . . . . 6 β’ (π β (π β π΄ β¦ π΅) = (π β π΄ β¦ π΅)) | |
33 | 32, 15 | fmpt3d 7069 | . . . . 5 β’ (π β (π β π΄ β¦ π΅):π΄βΆ(0[,]+β)) |
34 | 1, 2, 3, 15 | esumel 32686 | . . . . 5 β’ (π β Ξ£*π β π΄π΅ β ((β*π βΎs (0[,]+β)) tsums (π β π΄ β¦ π΅))) |
35 | 6, 9, 9, 18, 19, 18, 19, 31, 4, 3, 33, 34 | tsmsmhm 23513 | . . . 4 β’ (π β (πΆβΞ£*π β π΄π΅) β ((β*π βΎs (0[,]+β)) tsums (πΆ β (π β π΄ β¦ π΅)))) |
36 | 13, 15 | cofmpt 7083 | . . . . 5 β’ (π β (πΆ β (π β π΄ β¦ π΅)) = (π β π΄ β¦ (πΆβπ΅))) |
37 | 36 | oveq2d 7378 | . . . 4 β’ (π β ((β*π βΎs (0[,]+β)) tsums (πΆ β (π β π΄ β¦ π΅))) = ((β*π βΎs (0[,]+β)) tsums (π β π΄ β¦ (πΆβπ΅)))) |
38 | 35, 37 | eleqtrd 2840 | . . 3 β’ (π β (πΆβΞ£*π β π΄π΅) β ((β*π βΎs (0[,]+β)) tsums (π β π΄ β¦ (πΆβπ΅)))) |
39 | 1, 2, 3, 16, 38 | esumid 32683 | . 2 β’ (π β Ξ£*π β π΄(πΆβπ΅) = (πΆβΞ£*π β π΄π΅)) |
40 | 39 | eqcomd 2743 | 1 β’ (π β (πΆβΞ£*π β π΄π΅) = Ξ£*π β π΄(πΆβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 βͺ cuni 4870 β¦ cmpt 5193 β ccom 5642 βΆwf 6497 βcfv 6501 (class class class)co 7362 0cc0 11058 +βcpnf 11193 β€ cle 11197 +π cxad 13038 [,]cicc 13274 βΎs cress 17119 βΎt crest 17309 TopOpenctopn 17310 ordTopcordt 17388 β*π cxrs 17389 Mndcmnd 18563 MndHom cmhm 18606 CMndccmn 19569 TopSpctps 22297 Cn ccn 22591 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-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 |
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-of 7622 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-inf 9386 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-q 12881 df-xadd 13041 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-seq 13914 df-hash 14238 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-tset 17159 df-ple 17160 df-ds 17162 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-ordt 17390 df-xrs 17391 df-mre 17473 df-mrc 17474 df-acs 17476 df-ps 18462 df-tsr 18463 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-cntz 19104 df-cmn 19571 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-ntr 22387 df-nei 22465 df-cn 22594 df-cnp 22595 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: esummulc1 32720 |
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