Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbvesum Structured version   Unicode version

Theorem cbvesum 26643
Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
Hypotheses
Ref Expression
cbvesum.1  |-  ( j  =  k  ->  B  =  C )
cbvesum.2  |-  F/_ k A
cbvesum.3  |-  F/_ j A
cbvesum.4  |-  F/_ k B
cbvesum.5  |-  F/_ j C
Assertion
Ref Expression
cbvesum  |- Σ* j  e.  A B  = Σ* k  e.  A C
Distinct variable group:    j, k
Allowed substitution hints:    A( j, k)    B( j, k)    C( j, k)

Proof of Theorem cbvesum
StepHypRef Expression
1 cbvesum.3 . . . . 5  |-  F/_ j A
2 cbvesum.2 . . . . 5  |-  F/_ k A
3 cbvesum.4 . . . . 5  |-  F/_ k B
4 cbvesum.5 . . . . 5  |-  F/_ j C
5 cbvesum.1 . . . . 5  |-  ( j  =  k  ->  B  =  C )
61, 2, 3, 4, 5cbvmptf 26123 . . . 4  |-  ( j  e.  A  |->  B )  =  ( k  e.  A  |->  C )
76oveq2i 6212 . . 3  |-  ( (
RR*ss  ( 0 [,] +oo ) ) tsums  ( j  e.  A  |->  B ) )  =  ( (
RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
87unieqi 4209 . 2  |-  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( j  e.  A  |->  B ) )  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
9 df-esum 26630 . 2  |- Σ* j  e.  A B  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( j  e.  A  |->  B ) )
10 df-esum 26630 . 2  |- Σ* k  e.  A C  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
118, 9, 103eqtr4i 2493 1  |- Σ* j  e.  A B  = Σ* k  e.  A C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   F/_wnfc 2602   U.cuni 4200    |-> cmpt 4459  (class class class)co 6201   0cc0 9394   +oocpnf 9527   [,]cicc 11415   ↾s cress 14294   RR*scxrs 14558   tsums ctsu 19829  Σ*cesum 26629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-iota 5490  df-fv 5535  df-ov 6204  df-esum 26630
This theorem is referenced by:  cbvesumv  26644  esumfzf  26664
  Copyright terms: Public domain W3C validator