Theorem cbvsumv 14426

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 = k -> B = C )

Assertion

Ref	Expression
cbvsumv	|- sum_ j e. A B = sum_ k e. A C

Distinct variable groups:   j, k, A   B, k   C, j

Allowed substitution hints:   B( j)   C( k)

Proof of Theorem cbvsumv

Step	Hyp	Ref	Expression
1		cbvsum.1	. 2 |- ( j = k -> B = C )
2		nfcv 2764	. 2 |- F/_ k A
3		nfcv 2764	. 2 |- F/_ j A
4		nfcv 2764	. 2 |- F/_ k B
5		nfcv 2764	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvsum 14425	1 |- sum_ j e. A B = sum_ k e. A C

Syntax hints:   -> wi 4   = wceq 1483   sum_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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-sum 14417

This theorem is referenced by:  isumge0  14497  telfsumo  14534  fsumparts  14538  binomlem  14561  incexclem  14568  mertenslem1  14616  mertens  14618  binomfallfaclem2  14771  bpolyval  14780  efaddlem  14823  pwp1fsum  15114  bitsinv2  15165  prmreclem6  15625  ovolicc2lem4  23288  uniioombllem6  23356  plymullem1  23970  plyadd  23973  plymul  23974  coeeu  23981  coeid  23994  dvply1  24039  vieta1  24067  aaliou3  24106  abelthlem8  24193  abelthlem9  24194  abelth  24195  logtayl  24406  ftalem2  24800  ftalem6  24804  dchrsum2  24993  sumdchr2  24995  dchrisumlem1  25178  dchrisum  25181  dchrisum0fval  25194  dchrisum0ff  25196  rpvmasum  25215  mulog2sumlem1  25223  2vmadivsumlem  25229  logsqvma  25231  logsqvma2  25232  selberg  25237  chpdifbndlem1  25242  selberg3lem1  25246  selberg4lem1  25249  pntsval  25261  pntsval2  25265  pntpbnd1  25275  pntlemo  25296  axsegconlem9  25805  hashunif  29562  eulerpartlems  30422  eulerpartlemgs2  30442  breprexplema  30708  breprexplemc  30710  breprexp  30711  hgt750lema  30735  fwddifnp1  32272  binomcxplemnotnn0  38555  mccl  39830  sumnnodd  39862  dvnprodlem1  40161  dvnprodlem3  40163  dvnprod  40164  fourierdlem73  40396  fourierdlem112  40435  fourierdlem113  40436  etransclem11  40462  etransclem32  40483  etransclem35  40486  etransc  40500  fsumlesge0  40594  meaiuninclem  40697  omeiunltfirp  40733  hoidmvlelem3  40811  pwdif  41501  altgsumbcALT  42131  nn0sumshdiglemA  42413  nn0sumshdiglemB  42414