Theorem esumeq12dvaf 30093

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)

Hypotheses

Ref	Expression
esumeq12dvaf.1	|- F/ k ph
esumeq12dvaf.2	|- ( ph -> A = B )
esumeq12dvaf.3	|- ( ( ph /\ k e. A ) -> C = D )

Assertion

Ref	Expression
esumeq12dvaf	|- ( ph -> Σ* k e. A C = Σ* k e. B D )

Proof of Theorem esumeq12dvaf

Step	Hyp	Ref	Expression
1		esumeq12dvaf.1	. . . . . 6 |- F/ k ph
2		esumeq12dvaf.2	. . . . . 6 |- ( ph -> A = B )
3	1, 2	alrimi 2082	. . . . 5 |- ( ph -> A. k A = B )
4		esumeq12dvaf.3	. . . . . . 7 |- ( ( ph /\ k e. A ) -> C = D )
5	4	ex 450	. . . . . 6 |- ( ph -> ( k e. A -> C = D ) )
6	1, 5	ralrimi 2957	. . . . 5 |- ( ph -> A. k e. A C = D )
7		mpteq12f 4731	. . . . 5 |- ( ( A. k A = B /\ A. k e. A C = D ) -> ( k e. A |-> C ) = ( k e. B |-> D ) )
8	3, 6, 7	syl2anc 693	. . . 4 |- ( ph -> ( k e. A |-> C ) = ( k e. B |-> D ) )
9	8	oveq2d 6666	. . 3 |- ( ph -> ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. B |-> D ) ) )
10	9	unieqd 4446	. 2 |- ( ph -> U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) ) = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. B |-> D ) ) )
11		df-esum 30090	. 2 |- Σ* k e. A C = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) )
12		df-esum 30090	. 2 |- Σ* k e. B D = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. B |-> D ) )
13	10, 11, 12	3eqtr4g 2681	1 |- ( ph -> Σ* k e. A C = Σ* k e. B D )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   F/wnf 1708   e. wcel 1990   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:  esumeq12dva  30094  esumeq1d  30097  esumeq2d  30099  esumpinfval  30135  measvunilem0  30276