Theorem esumeq2 28265

Description: Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)

Assertion

Ref	Expression
esumeq2	|- ( A. k e. A B = C -> Σ* k e. A B = Σ* k e. A C )

Distinct variable group:   A, k

Allowed substitution hints:   B( k)   C( k)

Proof of Theorem esumeq2

Step	Hyp	Ref	Expression
1		eqid 2454	. . . . 5 |- A = A
2		mpteq12 4518	. . . . 5 |- ( ( A = A /\ A. k e. A B = C ) -> ( k e. A |-> B ) = ( k e. A |-> C ) )
3	1, 2	mpan 668	. . . 4 |- ( A. k e. A B = C -> ( k e. A |-> B ) = ( k e. A |-> C ) )
4	3	oveq2d 6286	. . 3 |- ( A. k e. A B = C -> ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) ) )
5	4	unieqd 4245	. 2 |- ( A. k e. A B = C -> U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) ) )
6		df-esum 28257	. 2 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
7		df-esum 28257	. 2 |- Σ* k e. A C = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) )
8	5, 6, 7	3eqtr4g 2520	1 |- ( A. k e. A B = C -> Σ* k e. A B = Σ* k e. A C )

Syntax hints:   -> wi 4   = wceq 1398   A.wral 2804   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: (None)