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Mirrors > Home > MPE Home > Th. List > sumex | Structured version Visualization version Unicode version |
Description: A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Ref | Expression |
---|---|
sumex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sum 14417 | . 2 | |
2 | iotaex 5868 | . 2 | |
3 | 1, 2 | eqeltri 2697 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wo 383 wa 384 wceq 1483 wex 1704 wcel 1990 wrex 2913 cvv 3200 csb 3533 wss 3574 cif 4086 class class class wbr 4653 cmpt 4729 cio 5849 wf1o 5887 cfv 5888 (class class class)co 6650 cc0 9936 c1 9937 caddc 9939 cn 11020 cz 11377 cuz 11687 cfz 12326 cseq 12801 cli 14215 csu 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-uni 4437 df-iota 5851 df-sum 14417 |
This theorem is referenced by: fsumrlim 14543 fsumo1 14544 efval 14810 efcvgfsum 14816 eftlub 14839 bitsinv2 15165 bitsinv 15170 lebnumlem3 22762 isi1f 23441 itg1val 23450 itg1climres 23481 itgex 23537 itgfsum 23593 dvmptfsum 23738 plyeq0lem 23966 plyaddlem1 23969 plymullem1 23970 coeeulem 23980 coeid2 23995 plyco 23997 coemullem 24006 coemul 24008 aareccl 24081 aaliou3lem5 24102 aaliou3lem6 24103 aaliou3lem7 24104 taylpval 24121 psercn 24180 pserdvlem2 24182 pserdv 24183 abelthlem6 24190 abelthlem8 24193 abelthlem9 24194 logtayl 24406 leibpi 24669 basellem3 24809 chtval 24836 chpval 24848 sgmval 24868 muinv 24919 dchrvmasumlem1 25184 dchrisum0fval 25194 dchrisum0fno1 25200 dchrisum0lem3 25208 dchrisum0 25209 mulogsum 25221 logsqvma2 25232 selberglem1 25234 pntsval 25261 ecgrtg 25863 esumpcvgval 30140 esumcvg 30148 eulerpartlemsv1 30418 signsplypnf 30627 signsvvfval 30655 vtsval 30715 circlemeth 30718 fwddifnval 32270 knoppndvlem6 32508 binomcxplemnotnn0 38555 stoweidlem11 40228 stoweidlem26 40243 fourierdlem112 40435 fsumlesge0 40594 sge0sn 40596 sge0f1o 40599 sge0supre 40606 sge0resplit 40623 sge0reuz 40664 sge0reuzb 40665 aacllem 42547 |
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