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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 21328. (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 21244 | . . 3 β’ β = (Baseββfld) | |
2 | cnfldadd 21246 | . . 3 β’ + = (+gββfld) | |
3 | eqid 2726 | . . 3 β’ (βfld βΎs β) = (βfld βΎs β) | |
4 | cnfldex 21243 | . . . 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 11169 | . . . 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 7110 | . . 3 β’ (π β (π β π΄ β¦ π΅):π΄βΆβ) |
11 | 0red 11221 | . . 3 β’ (π β 0 β β) | |
12 | simpr 484 | . . . . 5 β’ ((π β§ π₯ β β) β π₯ β β) | |
13 | 12 | addlidd 11419 | . . . 4 β’ ((π β§ π₯ β β) β (0 + π₯) = π₯) |
14 | 12 | addridd 11418 | . . . 4 β’ ((π β§ π₯ β β) β (π₯ + 0) = π₯) |
15 | 13, 14 | jca 511 | . . 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 18615 | . 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 11246 | . . 3 β’ ((π β§ π β π΄) β π΅ β β) |
18 | 6, 17 | gsumfsum 21328 | . 2 β’ (π β (βfld Ξ£g (π β π΄ β¦ π΅)) = Ξ£π β π΄ π΅) |
19 | 16, 18 | eqtr3d 2768 | 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 395 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 β¦ cmpt 5224 (class class class)co 7405 Fincfn 8941 βcc 11110 βcr 11111 0cc0 11112 + caddc 11115 Ξ£csu 15638 βΎs cress 17182 Ξ£g cgsu 17395 βfldccnfld 21240 |
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-inf2 9638 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 ax-addf 11191 |
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-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-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-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 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-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 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-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-0g 17396 df-gsum 17397 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-ur 20087 df-ring 20140 df-cring 20141 df-cnfld 21241 |
This theorem is referenced by: rrxdsfi 25294 |
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