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

Theorem esumeq12dvaf 26627
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 1815 . . . . 5  |-  ( ph  ->  A. k  A  =  B )
4 esumeq12dvaf.3 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  C  =  D )
54ex 434 . . . . . 6  |-  ( ph  ->  ( k  e.  A  ->  C  =  D ) )
61, 5ralrimi 2820 . . . . 5  |-  ( ph  ->  A. k  e.  A  C  =  D )
7 mpteq12f 4471 . . . . 5  |-  ( ( A. k  A  =  B  /\  A. k  e.  A  C  =  D )  ->  (
k  e.  A  |->  C )  =  ( k  e.  B  |->  D ) )
83, 6, 7syl2anc 661 . . . 4  |-  ( ph  ->  ( k  e.  A  |->  C )  =  ( k  e.  B  |->  D ) )
98oveq2d 6211 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e.  A  |->  C ) )  =  ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  B  |->  D ) ) )
109unieqd 4204 . 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 26624 . 2  |- Σ* k  e.  A C  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
12 df-esum 26624 . 2  |- Σ* k  e.  B D  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  B  |->  D ) )
1310, 11, 123eqtr4g 2518 1  |-  ( ph  -> Σ* k  e.  A C  = Σ* k  e.  B D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368    = wceq 1370   F/wnf 1590    e. wcel 1758   A.wral 2796   U.cuni 4194    |-> cmpt 4453  (class class class)co 6195   0cc0 9388   +oocpnf 9521   [,]cicc 11409   ↾s cress 14288   RR*scxrs 14552   tsums ctsu 19823  Σ*cesum 26623
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 1954  ax-ext 2431
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 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-iota 5484  df-fv 5529  df-ov 6198  df-esum 26624
This theorem is referenced by:  esumeq12dva  26628  esumeq1d  26631  esumeq2d  26633  esumpinfval  26662  measvunilem0  26767
  Copyright terms: Public domain W3C validator