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Mirrors > Home > MPE Home > Th. List > regsumfsum | Structured version Visualization version GIF version |
Description: Relate a group sum on (βfld βΎs β) to a finite sum on the reals. Cf. gsumfsum 20710. (Contributed by Thierry Arnoux, 7-Sep-2018.) |
Ref | Expression |
---|---|
regsumfsum.1 | β’ (π β π΄ β Fin) |
regsumfsum.2 | β’ ((π β§ π β π΄) β π΅ β β) |
Ref | Expression |
---|---|
regsumfsum | β’ (π β ((βfld βΎs β) Ξ£g (π β π΄ β¦ π΅)) = Ξ£π β π΄ π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 20646 | . . 3 β’ β = (Baseββfld) | |
2 | cnfldadd 20647 | . . 3 β’ + = (+gββfld) | |
3 | eqid 2736 | . . 3 β’ (βfld βΎs β) = (βfld βΎs β) | |
4 | cnfldex 20645 | . . . 4 β’ βfld β V | |
5 | 4 | a1i 11 | . . 3 β’ (π β βfld β V) |
6 | regsumfsum.1 | . . 3 β’ (π β π΄ β Fin) | |
7 | ax-resscn 10974 | . . . 4 β’ β β β | |
8 | 7 | a1i 11 | . . 3 β’ (π β β β β) |
9 | regsumfsum.2 | . . . 4 β’ ((π β§ π β π΄) β π΅ β β) | |
10 | 9 | fmpttd 7021 | . . 3 β’ (π β (π β π΄ β¦ π΅):π΄βΆβ) |
11 | 0red 11024 | . . 3 β’ (π β 0 β β) | |
12 | simpr 486 | . . . . 5 β’ ((π β§ π₯ β β) β π₯ β β) | |
13 | 12 | addid2d 11222 | . . . 4 β’ ((π β§ π₯ β β) β (0 + π₯) = π₯) |
14 | 12 | addid1d 11221 | . . . 4 β’ ((π β§ π₯ β β) β (π₯ + 0) = π₯) |
15 | 13, 14 | jca 513 | . . 3 β’ ((π β§ π₯ β β) β ((0 + π₯) = π₯ β§ (π₯ + 0) = π₯)) |
16 | 1, 2, 3, 5, 6, 8, 10, 11, 15 | gsumress 18411 | . 2 β’ (π β (βfld Ξ£g (π β π΄ β¦ π΅)) = ((βfld βΎs β) Ξ£g (π β π΄ β¦ π΅))) |
17 | 9 | recnd 11049 | . . 3 β’ ((π β§ π β π΄) β π΅ β β) |
18 | 6, 17 | gsumfsum 20710 | . 2 β’ (π β (βfld Ξ£g (π β π΄ β¦ π΅)) = Ξ£π β π΄ π΅) |
19 | 16, 18 | eqtr3d 2778 | 1 β’ (π β ((βfld βΎs β) Ξ£g (π β π΄ β¦ π΅)) = Ξ£π β π΄ π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1539 β wcel 2104 Vcvv 3437 β wss 3892 β¦ cmpt 5164 (class class class)co 7307 Fincfn 8764 βcc 10915 βcr 10916 0cc0 10917 + caddc 10920 Ξ£csu 15442 βΎs cress 16986 Ξ£g cgsu 17196 βfldccnfld 20642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-inf2 9443 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 ax-addf 10996 ax-mulf 10997 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-sup 9245 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-rp 12777 df-fz 13286 df-fzo 13429 df-seq 13768 df-exp 13829 df-hash 14091 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-clim 15242 df-sum 15443 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-starv 17022 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-0g 17197 df-gsum 17198 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-grp 18625 df-minusg 18626 df-cntz 18968 df-cmn 19433 df-abl 19434 df-mgp 19766 df-ur 19783 df-ring 19830 df-cring 19831 df-cnfld 20643 |
This theorem is referenced by: rrxdsfi 24620 |
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