Theorem esumex 30091

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 30090	. 2 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
2		ovex 6678	. . 3 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) e. _V
3	2	uniex 6953	. 2 |- U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) e. _V
4	1, 3	eqeltri 2697	1 |- Σ* k e. A B e. _V

Syntax hints:   e. wcel 1990   _Vcvv 3200   U.cuni 4436   |-> cmpt 4729  (class class class)co 6650   0cc0 9936   +oocpnf 10071   [,]cicc 12178   ↾_s cress 15858   RR*scxrs 16160   tsums ctsu 21929  Σ^*cesum 30089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896  df-ov 6653  df-esum 30090

This theorem is referenced by:  esumcvg  30148  esumgect  30152  omssubaddlem  30361  omssubadd  30362