Theorem esumeq2 26630

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 2451	. . . . 5 |- A = A
2		mpteq12 4472	. . . . 5 |- ( ( A = A /\ A. k e. A B = C ) -> ( k e. A |-> B ) = ( k e. A |-> C ) )
3	1, 2	mpan 670	. . . 4 |- ( A. k e. A B = C -> ( k e. A |-> B ) = ( k e. A |-> C ) )
4	3	oveq2d 6209	. . 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 4202	. 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 26622	. 2 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
7		df-esum 26622	. 2 |- Σ* k e. A C = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) )
8	5, 6, 7	3eqtr4g 2517	1 |- ( A. k e. A B = C -> Σ* k e. A B = Σ* k e. A C )

Syntax hints:   -> wi 4   = wceq 1370   A.wral 2795   U.cuni 4192   |-> cmpt 4451  (class class class)co 6193   0cc0 9386   +oocpnf 9519   [,]cicc 11407   ↾_s cress 14286   RR*scxrs 14549   tsums ctsu 19821  Σ^*cesum 26621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-iota 5482  df-fv 5527  df-ov 6196  df-esum 26622

This theorem is referenced by:  ddemeas  26789