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Theorem nfesum2 28270
Description: Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 2-May-2020.)
Hypotheses
Ref Expression
nfesum2.1  |-  F/_ x A
nfesum2.2  |-  F/_ x B
Assertion
Ref Expression
nfesum2  |-  F/_ xΣ* k  e.  A B
Distinct variable group:    x, k
Allowed substitution hints:    A( x, k)    B( x, k)

Proof of Theorem nfesum2
StepHypRef Expression
1 df-esum 28257 . 2  |- Σ* k  e.  A B  =  U. (
( RR*ss  ( 0 [,] +oo ) ) tsums  ( k  e.  A  |->  B ) )
2 nfcv 2616 . . . 4  |-  F/_ x
( RR*ss  ( 0 [,] +oo ) )
3 nfcv 2616 . . . 4  |-  F/_ x tsums
4 nfesum2.1 . . . . 5  |-  F/_ x A
5 nfesum2.2 . . . . 5  |-  F/_ x B
64, 5nfmpt 4527 . . . 4  |-  F/_ x
( k  e.  A  |->  B )
72, 3, 6nfov 6296 . . 3  |-  F/_ x
( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e.  A  |->  B ) )
87nfuni 4241 . 2  |-  F/_ x U. ( ( RR*ss  (
0 [,] +oo )
) tsums  ( k  e.  A  |->  B ) )
91, 8nfcxfr 2614 1  |-  F/_ xΣ* k  e.  A B
Colors of variables: wff setvar class
Syntax hints:   F/_wnfc 2602   U.cuni 4235    |-> cmpt 4497  (class class class)co 6270   0cc0 9481   +oocpnf 9614   [,]cicc 11535   ↾s cress 14717   RR*scxrs 14989   tsums ctsu 20790  Σ*cesum 28256
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 28257
This theorem is referenced by:  esum2dlem  28321
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