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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 2979 | . 2 ⊢ Ⅎ𝑘𝐵 | |
2 | fsum1.1 | . 2 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐵) | |
3 | 1, 2 | sumsnf 15101 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ ℂ) → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4569 ℂcc 10537 Σcsu 15044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 |
This theorem is referenced by: fsum1 15104 sumpr 15105 sumtp 15106 sumsns 15107 fsumm1 15108 fsum1p 15110 fsum2dlem 15127 fsumge1 15154 fsumrlim 15168 fsumo1 15169 fsumiun 15178 incexclem 15193 incexc 15194 binomfallfac 15397 fprodefsum 15450 rpnnen2lem11 15579 bitsinv1 15793 2ebits 15798 bitsinvp1 15800 ehl1eudis 24025 ovolfiniun 24104 volfiniun 24150 itg11 24294 itgfsum 24429 plyeq0lem 24802 coemulhi 24846 vieta1lem2 24902 vieta1 24903 chtprm 25732 musumsum 25771 muinv 25772 logexprlim 25803 perfectlem2 25808 dchrhash 25849 rpvmasum2 26090 eulerpartlems 31620 eulerpartlemgc 31622 plymulx0 31819 signsplypnf 31822 reprinfz1 31895 breprexp 31906 circlemeth 31913 ismrer1 35118 jm2.23 39600 k0004val0 40511 dvnprodlem3 42240 stoweidlem17 42309 stoweidlem44 42336 sge0cl 42670 carageniuncllem1 42810 perfectALTVlem2 43894 nnsum3primesprm 43962 nn0sumshdiglemB 44687 nn0sumshdiglem1 44688 nn0sumshdiglem2 44689 |
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