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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 2974 | . 2 ⊢ Ⅎ𝑘𝐵 | |
2 | fsum1.1 | . 2 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
3 | 1, 2 | sumsnf 15087 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 ℂcc 10523 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 |
This theorem is referenced by: fsum1 15090 sumpr 15091 sumtp 15092 sumsns 15093 fsumm1 15094 fsum1p 15096 fsum2dlem 15113 fsumge1 15140 fsumrlim 15154 fsumo1 15155 fsumiun 15164 incexclem 15179 incexc 15180 binomfallfac 15383 fprodefsum 15436 rpnnen2lem11 15565 bitsinv1 15779 2ebits 15784 bitsinvp1 15786 ehl1eudis 23950 ovolfiniun 24029 volfiniun 24075 itg11 24219 itgfsum 24354 plyeq0lem 24727 coemulhi 24771 vieta1lem2 24827 vieta1 24828 chtprm 25657 musumsum 25696 muinv 25697 logexprlim 25728 perfectlem2 25733 dchrhash 25774 rpvmasum2 26015 eulerpartlems 31517 eulerpartlemgc 31519 plymulx0 31716 signsplypnf 31719 reprinfz1 31792 breprexp 31803 circlemeth 31810 ismrer1 34997 jm2.23 39471 k0004val0 40382 dvnprodlem3 42109 stoweidlem17 42179 stoweidlem44 42206 sge0cl 42540 carageniuncllem1 42680 perfectALTVlem2 43764 nnsum3primesprm 43832 nn0sumshdiglemB 44608 nn0sumshdiglem1 44609 nn0sumshdiglem2 44610 |
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