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Mirrors > Home > MPE Home > Th. List > sumex | Structured version Visualization version GIF version |
Description: A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Ref | Expression |
---|---|
sumex | ⊢ Σ𝑘 ∈ 𝐴 𝐵 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sum 15719 | . 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) | |
2 | iotaex 6535 | . 2 ⊢ (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) ∈ V | |
3 | 1, 2 | eqeltri 2834 | 1 ⊢ Σ𝑘 ∈ 𝐴 𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 847 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ∃wrex 3067 Vcvv 3477 ⦋csb 3907 ⊆ wss 3962 ifcif 4530 class class class wbr 5147 ↦ cmpt 5230 ℩cio 6513 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 ℕcn 12263 ℤcz 12610 ℤ≥cuz 12875 ...cfz 13543 seqcseq 14038 ⇝ cli 15516 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-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-sn 4631 df-pr 4633 df-uni 4912 df-iota 6515 df-sum 15719 |
This theorem is referenced by: fsumrlim 15843 fsumo1 15844 efval 16111 efcvgfsum 16118 eftlub 16141 bitsinv2 16476 bitsinv 16481 lebnumlem3 25008 isi1f 25722 itg1val 25731 itg1climres 25763 itgex 25819 itgfsum 25876 dvmptfsum 26027 plyeq0lem 26263 plyaddlem1 26266 plymullem1 26267 coeeulem 26277 coeid2 26292 plyco 26294 coemullem 26303 coemul 26305 aareccl 26382 aaliou3lem5 26403 aaliou3lem6 26404 aaliou3lem7 26405 taylpval 26422 psercn 26484 pserdvlem2 26486 pserdv 26487 abelthlem6 26494 abelthlem8 26497 abelthlem9 26498 logtayl 26716 leibpi 26999 basellem3 27140 chtval 27167 chpval 27179 sgmval 27199 muinv 27250 dchrvmasumlem1 27553 dchrisum0fval 27563 dchrisum0fno1 27569 dchrisum0lem3 27577 dchrisum0 27578 mulogsum 27590 logsqvma2 27601 selberglem1 27603 pntsval 27630 ecgrtg 29012 esumpcvgval 34058 esumcvg 34066 eulerpartlemsv1 34337 signsplypnf 34543 signsvvfval 34571 vtsval 34630 circlemeth 34633 fwddifnval 36144 knoppndvlem6 36499 binomcxplemnotnn0 44351 stoweidlem11 45966 stoweidlem26 45981 fourierdlem112 46173 fsumlesge0 46332 sge0sn 46334 sge0f1o 46337 sge0supre 46344 sge0resplit 46361 sge0reuz 46402 sge0reuzb 46403 aacllem 49031 |
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