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Mirrors > Home > MPE Home > Th. List > fsummulc1 | Structured version Visualization version GIF version |
Description: A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsummulc2.1 | โข (๐ โ ๐ด โ Fin) |
fsummulc2.2 | โข (๐ โ ๐ถ โ โ) |
fsummulc2.3 | โข ((๐ โง ๐ โ ๐ด) โ ๐ต โ โ) |
Ref | Expression |
---|---|
fsummulc1 | โข (๐ โ (ฮฃ๐ โ ๐ด ๐ต ยท ๐ถ) = ฮฃ๐ โ ๐ด (๐ต ยท ๐ถ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsummulc2.1 | . . 3 โข (๐ โ ๐ด โ Fin) | |
2 | fsummulc2.2 | . . 3 โข (๐ โ ๐ถ โ โ) | |
3 | fsummulc2.3 | . . 3 โข ((๐ โง ๐ โ ๐ด) โ ๐ต โ โ) | |
4 | 1, 2, 3 | fsummulc2 15734 | . 2 โข (๐ โ (๐ถ ยท ฮฃ๐ โ ๐ด ๐ต) = ฮฃ๐ โ ๐ด (๐ถ ยท ๐ต)) |
5 | 1, 3 | fsumcl 15683 | . . 3 โข (๐ โ ฮฃ๐ โ ๐ด ๐ต โ โ) |
6 | 5, 2 | mulcomd 11239 | . 2 โข (๐ โ (ฮฃ๐ โ ๐ด ๐ต ยท ๐ถ) = (๐ถ ยท ฮฃ๐ โ ๐ด ๐ต)) |
7 | 2 | adantr 479 | . . . 4 โข ((๐ โง ๐ โ ๐ด) โ ๐ถ โ โ) |
8 | 3, 7 | mulcomd 11239 | . . 3 โข ((๐ โง ๐ โ ๐ด) โ (๐ต ยท ๐ถ) = (๐ถ ยท ๐ต)) |
9 | 8 | sumeq2dv 15653 | . 2 โข (๐ โ ฮฃ๐ โ ๐ด (๐ต ยท ๐ถ) = ฮฃ๐ โ ๐ด (๐ถ ยท ๐ต)) |
10 | 4, 6, 9 | 3eqtr4d 2780 | 1 โข (๐ โ (ฮฃ๐ โ ๐ด ๐ต ยท ๐ถ) = ฮฃ๐ โ ๐ด (๐ต ยท ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 394 = wceq 1539 โ wcel 2104 (class class class)co 7411 Fincfn 8941 โcc 11110 ยท cmul 11117 ฮฃcsu 15636 |
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 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 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 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 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-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 |
This theorem is referenced by: fsumdivc 15736 fsum2mul 15739 binomlem 15779 geoserg 15816 geo2sum 15823 mertenslem1 15834 binomfallfaclem2 15988 csbren 25147 plymullem1 25963 aalioulem1 26081 aaliou3lem6 26097 ftalem1 26813 ftalem5 26817 musumsum 26932 muinv 26933 fsumdvdsmul 26935 vmadivsum 27221 dchrisumlem2 27229 dchrmusum2 27233 dchrvmasumiflem2 27241 rpvmasum2 27251 dchrisum0lem1 27255 dchrisum0lem2a 27256 mulogsumlem 27270 mulog2sumlem3 27275 vmalogdivsum 27278 2vmadivsumlem 27279 logsqvma 27281 selberg3lem1 27296 selberg4 27300 pntrlog2bndlem5 27320 eulerpartlemgs2 33677 breprexplemc 33942 breprexpnat 33944 circlemeth 33950 hgt750lemb 33966 aks4d1p1p1 41234 jm2.23 42037 fsummulc1f 44585 dvnprodlem2 44961 dirkertrigeqlem2 45113 etransclem23 45271 etransclem46 45294 hoidmvlelem2 45610 nn0sumshdiglemA 47392 nn0sumshdiglemB 47393 nn0mullong 47398 aacllem 47935 amgmlemALT 47937 |
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