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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcocn | Structured version Visualization version GIF version |
Description: Lemma for esummulc2 33734 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 1909 | . . 3 β’ β²ππ | |
2 | nfcv 2899 | . . 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 33576 | . . . . . . . 8 β’ (β*π βΎs (0[,]+β)) β TopSp | |
6 | xrge0base 32762 | . . . . . . . . 9 β’ (0[,]+β) = (Baseβ(β*π βΎs (0[,]+β))) | |
7 | esumcocn.j | . . . . . . . . . 10 β’ π½ = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
8 | xrge0topn 33577 | . . . . . . . . . 10 β’ (TopOpenβ(β*π βΎs (0[,]+β))) = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
9 | 7, 8 | eqtr4i 2759 | . . . . . . . . 9 β’ π½ = (TopOpenβ(β*π βΎs (0[,]+β))) |
10 | 6, 9 | tpsuni 22858 | . . . . . . . 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 23170 | . . . . . 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 479 | . . . 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 7100 | . . 3 β’ ((π β§ π β π΄) β (πΆβπ΅) β (0[,]+β)) |
17 | xrge0cmn 21348 | . . . . . 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 19759 | . . . . . . . 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 1119 | . . . . . . 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 32764 | . . . . . . . 8 β’ +π = (+gβ(β*π βΎs (0[,]+β))) | |
28 | xrge00 32763 | . . . . . . . 8 β’ 0 = (0gβ(β*π βΎs (0[,]+β))) | |
29 | 6, 6, 27, 27, 28, 28 | ismhm 18749 | . . . . . . 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 1374 | . . . . 5 β’ (π β πΆ β ((β*π βΎs (0[,]+β)) MndHom (β*π βΎs (0[,]+β)))) |
32 | eqidd 2729 | . . . . . 6 β’ (π β (π β π΄ β¦ π΅) = (π β π΄ β¦ π΅)) | |
33 | 32, 15 | fmpt3d 7131 | . . . . 5 β’ (π β (π β π΄ β¦ π΅):π΄βΆ(0[,]+β)) |
34 | 1, 2, 3, 15 | esumel 33699 | . . . . 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 24070 | . . . 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 7147 | . . . . 5 β’ (π β (πΆ β (π β π΄ β¦ π΅)) = (π β π΄ β¦ (πΆβπ΅))) |
37 | 36 | oveq2d 7442 | . . . 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 2831 | . . 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 33696 | . 2 β’ (π β Ξ£*π β π΄(πΆβπ΅) = (πΆβΞ£*π β π΄π΅)) |
40 | 39 | eqcomd 2734 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 βͺ cuni 4912 β¦ cmpt 5235 β ccom 5686 βΆwf 6549 βcfv 6553 (class class class)co 7426 0cc0 11146 +βcpnf 11283 β€ cle 11287 +π cxad 13130 [,]cicc 13367 βΎs cress 17216 βΎt crest 17409 TopOpenctopn 17410 ordTopcordt 17488 β*π cxrs 17489 Mndcmnd 18701 MndHom cmhm 18745 CMndccmn 19742 TopSpctps 22854 Cn ccn 23148 tsums ctsu 24050 Ξ£*cesum 33679 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 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 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-xadd 13133 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-tset 17259 df-ple 17260 df-ds 17262 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-ordt 17490 df-xrs 17491 df-mre 17573 df-mrc 17574 df-acs 17576 df-ps 18565 df-tsr 18566 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-cntz 19275 df-cmn 19744 df-fbas 21283 df-fg 21284 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-ntr 22944 df-nei 23022 df-cn 23151 df-cnp 23152 df-haus 23239 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-tsms 24051 df-esum 33680 |
This theorem is referenced by: esummulc1 33733 |
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