Theorem esumeq2 30098

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 2622	. . . . 5 |- A = A
2		mpteq12 4736	. . . . 5 |- ( ( A = A /\ A. k e. A B = C ) -> ( k e. A |-> B ) = ( k e. A |-> C ) )
3	1, 2	mpan 706	. . . 4 |- ( A. k e. A B = C -> ( k e. A |-> B ) = ( k e. A |-> C ) )
4	3	oveq2d 6666	. . 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 4446	. 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 30090	. 2 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
7		df-esum 30090	. 2 |- Σ* k e. A C = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) )
8	5, 6, 7	3eqtr4g 2681	1 |- ( A. k e. A B = C -> Σ* k e. A B = Σ* k e. A C )

Syntax hints:   -> wi 4   = wceq 1483   A.wral 2912   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-ov 6653  df-esum 30090

This theorem is referenced by: (None)