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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 15670 | . 2 โข (๐ โ (๐ถ ยท ฮฃ๐ โ ๐ด ๐ต) = ฮฃ๐ โ ๐ด (๐ถ ยท ๐ต)) |
5 | 1, 3 | fsumcl 15619 | . . 3 โข (๐ โ ฮฃ๐ โ ๐ด ๐ต โ โ) |
6 | 5, 2 | mulcomd 11177 | . 2 โข (๐ โ (ฮฃ๐ โ ๐ด ๐ต ยท ๐ถ) = (๐ถ ยท ฮฃ๐ โ ๐ด ๐ต)) |
7 | 2 | adantr 482 | . . . 4 โข ((๐ โง ๐ โ ๐ด) โ ๐ถ โ โ) |
8 | 3, 7 | mulcomd 11177 | . . 3 โข ((๐ โง ๐ โ ๐ด) โ (๐ต ยท ๐ถ) = (๐ถ ยท ๐ต)) |
9 | 8 | sumeq2dv 15589 | . 2 โข (๐ โ ฮฃ๐ โ ๐ด (๐ต ยท ๐ถ) = ฮฃ๐ โ ๐ด (๐ถ ยท ๐ต)) |
10 | 4, 6, 9 | 3eqtr4d 2787 | 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 397 = wceq 1542 โ wcel 2107 (class class class)co 7358 Fincfn 8884 โcc 11050 ยท cmul 11057 ฮฃcsu 15571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-oi 9447 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-fz 13426 df-fzo 13569 df-seq 13908 df-exp 13969 df-hash 14232 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 df-sum 15572 |
This theorem is referenced by: fsumdivc 15672 fsum2mul 15675 binomlem 15715 geoserg 15752 geo2sum 15759 mertenslem1 15770 binomfallfaclem2 15924 csbren 24766 plymullem1 25578 aalioulem1 25695 aaliou3lem6 25711 ftalem1 26425 ftalem5 26429 musumsum 26544 muinv 26545 fsumdvdsmul 26547 vmadivsum 26833 dchrisumlem2 26841 dchrmusum2 26845 dchrvmasumiflem2 26853 rpvmasum2 26863 dchrisum0lem1 26867 dchrisum0lem2a 26868 mulogsumlem 26882 mulog2sumlem3 26887 vmalogdivsum 26890 2vmadivsumlem 26891 logsqvma 26893 selberg3lem1 26908 selberg4 26912 pntrlog2bndlem5 26932 eulerpartlemgs2 32983 breprexplemc 33248 breprexpnat 33250 circlemeth 33256 hgt750lemb 33272 aks4d1p1p1 40523 jm2.23 41323 fsummulc1f 43819 dvnprodlem2 44195 dirkertrigeqlem2 44347 etransclem23 44505 etransclem46 44528 hoidmvlelem2 44844 nn0sumshdiglemA 46712 nn0sumshdiglemB 46713 nn0mullong 46718 aacllem 47255 amgmlemALT 47257 |
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