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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 2969 | . 2 ⊢ Ⅎ𝑘𝐵 | |
2 | fsum1.1 | . 2 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
3 | 1, 2 | sumsnf 14850 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 {csn 4397 ℂcc 10250 Σcsu 14793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-sum 14794 |
This theorem is referenced by: fsum1 14853 sumpr 14854 sumtp 14855 sumsns 14856 fsumm1 14857 fsum1p 14859 fsum2dlem 14876 fsumge1 14903 fsumrlim 14917 fsumo1 14918 fsumiun 14927 incexclem 14942 incexc 14943 binomfallfac 15144 fprodefsum 15197 rpnnen2lem11 15327 bitsinv1 15537 2ebits 15542 bitsinvp1 15544 ehl1eudis 23588 ovolfiniun 23667 volfiniun 23713 itg11 23857 itgfsum 23992 plyeq0lem 24365 coemulhi 24409 vieta1lem2 24465 vieta1 24466 chtprm 25292 musumsum 25331 muinv 25332 logexprlim 25363 perfectlem2 25368 dchrhash 25409 rpvmasum2 25614 eulerpartlems 30956 eulerpartlemgc 30958 plymulx0 31160 signsplypnf 31163 reprinfz1 31238 breprexp 31249 circlemeth 31256 ismrer1 34172 jm2.23 38399 k0004val0 39285 dvnprodlem3 40951 stoweidlem17 41021 stoweidlem44 41048 sge0cl 41382 carageniuncllem1 41522 perfectALTVlem2 42454 nnsum3primesprm 42501 nn0sumshdiglemB 43254 nn0sumshdiglem1 43255 nn0sumshdiglem2 43256 |
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