Theorem esumex 28258

Description: An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)

Assertion

Ref	Expression
esumex	|- Σ* k e. A B e. _V

Proof of Theorem esumex

Step	Hyp	Ref	Expression
1		df-esum 28257	. 2 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
2		ovex 6298	. . 3 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) e. _V
3	2	uniex 6569	. 2 |- U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) e. _V
4	1, 3	eqeltri 2538	1 |- Σ* k e. A B e. _V

Syntax hints:   e. wcel 1823   _Vcvv 3106   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-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019  df-uni 4236  df-iota 5534  df-fv 5578  df-ov 6273  df-esum 28257

This theorem is referenced by:  esumcvg  28315  esumgect  28319  omssubaddlem  28507  omssubadd  28508