MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvsumv Structured version   Visualization version   GIF version

Theorem cbvsumv 15041
Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Hypothesis
Ref Expression
cbvsum.1 (𝑗 = 𝑘𝐵 = 𝐶)
Assertion
Ref Expression
cbvsumv Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑘   𝐶,𝑗
Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)

Proof of Theorem cbvsumv
StepHypRef Expression
1 cbvsum.1 . 2 (𝑗 = 𝑘𝐵 = 𝐶)
2 nfcv 2974 . 2 𝑘𝐴
3 nfcv 2974 . 2 𝑗𝐴
4 nfcv 2974 . 2 𝑘𝐵
5 nfcv 2974 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvsum 15040 1 Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  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
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  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-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seq 13358  df-sum 15031
This theorem is referenced by:  isumge0  15109  telfsumo  15145  fsumparts  15149  binomlem  15172  incexclem  15179  pwdif  15211  mertenslem1  15228  mertens  15230  binomfallfaclem2  15382  bpolyval  15391  efaddlem  15434  pwp1fsum  15730  bitsinv2  15780  prmreclem6  16245  ovolicc2lem4  24048  uniioombllem6  24116  plymullem1  24731  plyadd  24734  plymul  24735  coeeu  24742  coeid  24755  dvply1  24800  vieta1  24828  aaliou3  24867  abelthlem8  24954  abelthlem9  24955  abelth  24956  logtayl  25170  ftalem2  25578  ftalem6  25582  dchrsum2  25771  sumdchr2  25773  dchrisumlem1  25992  dchrisum  25995  dchrisum0fval  26008  dchrisum0ff  26010  rpvmasum  26029  mulog2sumlem1  26037  2vmadivsumlem  26043  logsqvma  26045  logsqvma2  26046  selberg  26051  chpdifbndlem1  26056  selberg3lem1  26060  selberg4lem1  26063  pntsval  26075  pntsval2  26079  pntpbnd1  26089  pntlemo  26110  axsegconlem9  26638  hashunif  30454  eulerpartlems  31517  eulerpartlemgs2  31537  breprexplema  31800  breprexplemc  31802  breprexp  31803  hgt750lema  31827  fwddifnp1  33523  binomcxplemnotnn0  40565  mccl  41755  sumnnodd  41787  dvnprodlem1  42107  dvnprodlem3  42109  dvnprod  42110  fourierdlem73  42341  fourierdlem112  42380  fourierdlem113  42381  etransclem11  42407  etransclem32  42428  etransclem35  42431  etransc  42445  fsumlesge0  42536  meaiuninclem  42639  omeiunltfirp  42678  hoidmvlelem3  42756  altgsumbcALT  44329  nn0sumshdiglemA  44607  nn0sumshdiglemB  44608
  Copyright terms: Public domain W3C validator