Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumeq2 Structured version   Unicode version

Theorem esumeq2 28809
Description: Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
Assertion
Ref Expression
esumeq2  |-  ( A. k  e.  A  B  =  C  -> Σ* k  e.  A B  = Σ* k  e.  A C )
Distinct variable group:    A, k
Allowed substitution hints:    B( k)    C( k)

Proof of Theorem esumeq2
StepHypRef Expression
1 eqid 2428 . . . . 5  |-  A  =  A
2 mpteq12 4446 . . . . 5  |-  ( ( A  =  A  /\  A. k  e.  A  B  =  C )  ->  (
k  e.  A  |->  B )  =  ( k  e.  A  |->  C ) )
31, 2mpan 674 . . . 4  |-  ( A. k  e.  A  B  =  C  ->  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  C ) )
43oveq2d 6265 . . 3  |-  ( A. k  e.  A  B  =  C  ->  ( (
RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )  =  ( (
RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) ) )
54unieqd 4172 . 2  |-  ( A. k  e.  A  B  =  C  ->  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) ) )
6 df-esum 28801 . 2  |- Σ* k  e.  A B  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )
7 df-esum 28801 . 2  |- Σ* k  e.  A C  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
85, 6, 73eqtr4g 2487 1  |-  ( A. k  e.  A  B  =  C  -> Σ* k  e.  A B  = Σ* k  e.  A C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   A.wral 2714   U.cuni 4162    |-> cmpt 4425  (class class class)co 6249   0cc0 9490   +oocpnf 9623   [,]cicc 11589   ↾s cress 15065   RR*scxrs 15341   tsums ctsu 21082  Σ*cesum 28800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-iota 5508  df-fv 5552  df-ov 6252  df-esum 28801
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator