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Theorem esumeq12dvaf 27910
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 1861 . . . . 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 2841 . . . . 5  |-  ( ph  ->  A. k  e.  A  C  =  D )
7 mpteq12f 4509 . . . . 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 6293 . . 3  |-  ( ph  ->  ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e.  A  |->  C ) )  =  ( ( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  B  |->  D ) ) )
109unieqd 4240 . 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 27907 . 2  |- Σ* k  e.  A C  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  C ) )
12 df-esum 27907 . 2  |- Σ* k  e.  B D  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  B  |->  D ) )
1310, 11, 123eqtr4g 2507 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 1379    = wceq 1381   F/wnf 1601    e. wcel 1802   A.wral 2791   U.cuni 4230    |-> cmpt 4491  (class class class)co 6277   0cc0 9490   +oocpnf 9623   [,]cicc 11536   ↾s cress 14505   RR*scxrs 14769   tsums ctsu 20490  Σ*cesum 27906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-iota 5537  df-fv 5582  df-ov 6280  df-esum 27907
This theorem is referenced by:  esumeq12dva  27911  esumeq1d  27914  esumeq2d  27916  esumpinfval  27945  measvunilem0  28050
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