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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 15735 | . 2 โข (๐ โ (๐ถ ยท ฮฃ๐ โ ๐ด ๐ต) = ฮฃ๐ โ ๐ด (๐ถ ยท ๐ต)) |
5 | 1, 3 | fsumcl 15684 | . . 3 โข (๐ โ ฮฃ๐ โ ๐ด ๐ต โ โ) |
6 | 5, 2 | mulcomd 11240 | . 2 โข (๐ โ (ฮฃ๐ โ ๐ด ๐ต ยท ๐ถ) = (๐ถ ยท ฮฃ๐ โ ๐ด ๐ต)) |
7 | 2 | adantr 480 | . . . 4 โข ((๐ โง ๐ โ ๐ด) โ ๐ถ โ โ) |
8 | 3, 7 | mulcomd 11240 | . . 3 โข ((๐ โง ๐ โ ๐ด) โ (๐ต ยท ๐ถ) = (๐ถ ยท ๐ต)) |
9 | 8 | sumeq2dv 15654 | . 2 โข (๐ โ ฮฃ๐ โ ๐ด (๐ต ยท ๐ถ) = ฮฃ๐ โ ๐ด (๐ถ ยท ๐ต)) |
10 | 4, 6, 9 | 3eqtr4d 2781 | 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 395 = wceq 1540 โ wcel 2105 (class class class)co 7412 Fincfn 8942 โcc 11111 ยท cmul 11118 ฮฃcsu 15637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 |
This theorem is referenced by: fsumdivc 15737 fsum2mul 15740 binomlem 15780 geoserg 15817 geo2sum 15824 mertenslem1 15835 binomfallfaclem2 15989 csbren 25148 plymullem1 25961 aalioulem1 26078 aaliou3lem6 26094 ftalem1 26810 ftalem5 26814 musumsum 26929 muinv 26930 fsumdvdsmul 26932 vmadivsum 27218 dchrisumlem2 27226 dchrmusum2 27230 dchrvmasumiflem2 27238 rpvmasum2 27248 dchrisum0lem1 27252 dchrisum0lem2a 27253 mulogsumlem 27267 mulog2sumlem3 27272 vmalogdivsum 27275 2vmadivsumlem 27276 logsqvma 27278 selberg3lem1 27293 selberg4 27297 pntrlog2bndlem5 27317 eulerpartlemgs2 33674 breprexplemc 33939 breprexpnat 33941 circlemeth 33947 hgt750lemb 33963 aks4d1p1p1 41235 jm2.23 42038 fsummulc1f 44587 dvnprodlem2 44963 dirkertrigeqlem2 45115 etransclem23 45273 etransclem46 45296 hoidmvlelem2 45612 nn0sumshdiglemA 47394 nn0sumshdiglemB 47395 nn0mullong 47400 aacllem 47937 amgmlemALT 47939 |
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