Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  esumeq12dvaf Structured version   Unicode version

Theorem esumeq12dvaf 28263
Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
Hypotheses
Ref Expression
esumeq12dvaf.1  |-  F/ k
ph
esumeq12dvaf.2  |-  ( ph  ->  A  =  B )
esumeq12dvaf.3  |-  ( (
ph  /\  k  e.  A )  ->  C  =  D )
Assertion
Ref Expression
esumeq12dvaf  |-  ( ph  -> Σ* k  e.  A C  = Σ* k  e.  B D )

Proof of Theorem esumeq12dvaf
StepHypRef Expression
1 esumeq12dvaf.1 . . . . . 6  |-  F/ k
ph
2 esumeq12dvaf.2 . . . . . 6  |-  ( ph  ->  A  =  B )
31, 2alrimi 1882 . . . . 5  |-  ( ph  ->  A. k  A  =  B )
4 esumeq12dvaf.3 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  C  =  D )
54ex 432 . . . . . 6  |-  ( ph  ->  ( k  e.  A  ->  C  =  D ) )
61, 5ralrimi 2854 . . . . 5  |-  ( ph  ->  A. k  e.  A  C  =  D )
7 mpteq12f 4515 . . . . 5  |-  ( ( A. k  A  =  B  /\  A. k  e.  A  C  =  D )  ->  (
k  e.  A  |->  C )  =  ( k  e.  B  |->  D ) )
83, 6, 7syl2anc 659 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  C )  =  ( k  e.  B  |->  D ) )
98oveq2d 6286 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e.  A  |->  C ) )  =  ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  B  |->  D ) ) )
109unieqd 4245 . 2  |-  ( ph  ->  U. ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  B  |->  D ) ) )
11 df-esum 28260 . 2  |- Σ* k  e.  A C  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
12 df-esum 28260 . 2  |- Σ* k  e.  B D  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  B  |->  D ) )
1310, 11, 123eqtr4g 2520 1  |-  ( ph  -> Σ* k  e.  A C  = Σ* k  e.  B D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396    = wceq 1398   F/wnf 1621    e. wcel 1823   A.wral 2804   U.cuni 4235    |-> cmpt 4497  (class class class)co 6270   0cc0 9481   +oocpnf 9614   [,]cicc 11535   ↾s cress 14720   RR*scxrs 14992   tsums ctsu 20793  Σ*cesum 28259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-iota 5534  df-fv 5578  df-ov 6273  df-esum 28260
This theorem is referenced by:  esumeq12dva  28264  esumeq1d  28267  esumeq2d  28269  esumpinfval  28305  measvunilem0  28424
  Copyright terms: Public domain W3C validator