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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 2902 | . 2 ⊢ Ⅎ𝑘𝐵 | |
2 | fsum1.1 | . 2 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
3 | 1, 2 | sumsnf 15775 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {csn 4630 ℂcc 11150 Σcsu 15718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 |
This theorem is referenced by: fsum1 15779 sumpr 15780 sumtp 15781 sumsns 15782 fsumm1 15783 fsum1p 15785 fsum2dlem 15802 fsumge1 15829 fsumrlim 15843 fsumo1 15844 fsumiun 15853 incexclem 15868 incexc 15869 binomfallfac 16073 fprodefsum 16127 rpnnen2lem11 16256 bitsinv1 16475 2ebits 16480 bitsinvp1 16482 ehl1eudis 25467 ovolfiniun 25549 volfiniun 25595 itg11 25739 itgfsum 25876 plyeq0lem 26263 coemulhi 26307 vieta1lem2 26367 vieta1 26368 chtprm 27210 musumsum 27249 muinv 27250 logexprlim 27283 perfectlem2 27288 dchrhash 27329 rpvmasum2 27570 eulerpartlems 34341 eulerpartlemgc 34343 plymulx0 34540 signsplypnf 34543 reprinfz1 34615 breprexp 34626 circlemeth 34633 ismrer1 37824 sticksstones9 42135 sticksstones11 42137 jm2.23 42984 k0004val0 44143 dvnprodlem3 45903 stoweidlem17 45972 stoweidlem44 45999 sge0cl 46336 carageniuncllem1 46476 perfectALTVlem2 47646 nnsum3primesprm 47714 nn0sumshdiglemB 48469 nn0sumshdiglem1 48470 nn0sumshdiglem2 48471 |
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