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Mirrors > Home > MPE Home > Th. List > sumsn | Structured version Visualization version GIF version |
Description: A sum of a singleton is the term. (Contributed by Mario Carneiro, 22-Apr-2014.) |
Ref | Expression |
---|---|
fsum1.1 | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
sumsn | ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2900 | . 2 ⊢ Ⅎ𝑘𝐵 | |
2 | fsum1.1 | . 2 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
3 | 1, 2 | sumsnf 15195 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {csn 4517 ℂcc 10616 Σcsu 15138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-sup 8982 df-oi 9050 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-n0 11980 df-z 12066 df-uz 12328 df-rp 12476 df-fz 12985 df-fzo 13128 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-clim 14938 df-sum 15139 |
This theorem is referenced by: fsum1 15198 sumpr 15199 sumtp 15200 sumsns 15201 fsumm1 15202 fsum1p 15204 fsum2dlem 15221 fsumge1 15248 fsumrlim 15262 fsumo1 15263 fsumiun 15272 incexclem 15287 incexc 15288 binomfallfac 15490 fprodefsum 15543 rpnnen2lem11 15672 bitsinv1 15888 2ebits 15893 bitsinvp1 15895 ehl1eudis 24175 ovolfiniun 24256 volfiniun 24302 itg11 24446 itgfsum 24582 plyeq0lem 24962 coemulhi 25006 vieta1lem2 25062 vieta1 25063 chtprm 25893 musumsum 25932 muinv 25933 logexprlim 25964 perfectlem2 25969 dchrhash 26010 rpvmasum2 26251 eulerpartlems 31900 eulerpartlemgc 31902 plymulx0 32099 signsplypnf 32102 reprinfz1 32175 breprexp 32186 circlemeth 32193 ismrer1 35642 jm2.23 40413 k0004val0 41333 dvnprodlem3 43054 stoweidlem17 43123 stoweidlem44 43150 sge0cl 43484 carageniuncllem1 43624 perfectALTVlem2 44738 nnsum3primesprm 44806 nn0sumshdiglemB 45530 nn0sumshdiglem1 45531 nn0sumshdiglem2 45532 |
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