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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcocn | Structured version Visualization version GIF version |
Description: Lemma for esummulc2 33610 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 2897 | . . 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 33452 | . . . . . . . 8 β’ (β*π βΎs (0[,]+β)) β TopSp | |
6 | xrge0base 32689 | . . . . . . . . 9 β’ (0[,]+β) = (Baseβ(β*π βΎs (0[,]+β))) | |
7 | esumcocn.j | . . . . . . . . . 10 β’ π½ = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
8 | xrge0topn 33453 | . . . . . . . . . 10 β’ (TopOpenβ(β*π βΎs (0[,]+β))) = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
9 | 7, 8 | eqtr4i 2757 | . . . . . . . . 9 β’ π½ = (TopOpenβ(β*π βΎs (0[,]+β))) |
10 | 6, 9 | tpsuni 22793 | . . . . . . . 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 23105 | . . . . . 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 480 | . . . 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 7081 | . . 3 β’ ((π β§ π β π΄) β (πΆβπ΅) β (0[,]+β)) |
17 | xrge0cmn 21302 | . . . . . 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 19717 | . . . . . . . 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 3192 | . . . . . 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 32691 | . . . . . . . 8 β’ +π = (+gβ(β*π βΎs (0[,]+β))) | |
28 | xrge00 32690 | . . . . . . . 8 β’ 0 = (0gβ(β*π βΎs (0[,]+β))) | |
29 | 6, 6, 27, 27, 28, 28 | ismhm 18715 | . . . . . . 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 2727 | . . . . . 6 β’ (π β (π β π΄ β¦ π΅) = (π β π΄ β¦ π΅)) | |
33 | 32, 15 | fmpt3d 7111 | . . . . 5 β’ (π β (π β π΄ β¦ π΅):π΄βΆ(0[,]+β)) |
34 | 1, 2, 3, 15 | esumel 33575 | . . . . 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 24005 | . . . 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 7126 | . . . . 5 β’ (π β (πΆ β (π β π΄ β¦ π΅)) = (π β π΄ β¦ (πΆβπ΅))) |
37 | 36 | oveq2d 7421 | . . . 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 2829 | . . 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 33572 | . 2 β’ (π β Ξ£*π β π΄(πΆβπ΅) = (πΆβΞ£*π β π΄π΅)) |
40 | 39 | eqcomd 2732 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 βͺ cuni 4902 β¦ cmpt 5224 β ccom 5673 βΆwf 6533 βcfv 6537 (class class class)co 7405 0cc0 11112 +βcpnf 11249 β€ cle 11253 +π cxad 13096 [,]cicc 13333 βΎs cress 17182 βΎt crest 17375 TopOpenctopn 17376 ordTopcordt 17454 β*π cxrs 17455 Mndcmnd 18667 MndHom cmhm 18711 CMndccmn 19700 TopSpctps 22789 Cn ccn 23083 tsums ctsu 23985 Ξ£*cesum 33555 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-xadd 13099 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-tset 17225 df-ple 17226 df-ds 17228 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-ordt 17456 df-xrs 17457 df-mre 17539 df-mrc 17540 df-acs 17542 df-ps 18531 df-tsr 18532 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-cntz 19233 df-cmn 19702 df-fbas 21237 df-fg 21238 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-ntr 22879 df-nei 22957 df-cn 23086 df-cnp 23087 df-haus 23174 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-tsms 23986 df-esum 33556 |
This theorem is referenced by: esummulc1 33609 |
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