Theorem cbvesum 30104

Description: Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)

Hypotheses

Ref	Expression
cbvesum.1	|- ( j = k -> B = C )
cbvesum.2	|- F/_ k A
cbvesum.3	|- F/_ j A
cbvesum.4	|- F/_ k B
cbvesum.5	|- F/_ j C

Assertion

Ref	Expression
cbvesum	|- Σ* j e. A B = Σ* k e. A C

Distinct variable group:   j, k

Allowed substitution hints:   A( j, k)   B( j, k)   C( j, k)

Proof of Theorem cbvesum

Step	Hyp	Ref	Expression
1		cbvesum.3	. . . . 5 |- F/_ j A
2		cbvesum.2	. . . . 5 |- F/_ k A
3		cbvesum.4	. . . . 5 |- F/_ k B
4		cbvesum.5	. . . . 5 |- F/_ j C
5		cbvesum.1	. . . . 5 |- ( j = k -> B = C )
6	1, 2, 3, 4, 5	cbvmptf 4748	. . . 4 |- ( j e. A |-> B ) = ( k e. A |-> C )
7	6	oveq2i 6661	. . 3 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( j e. A |-> B ) ) = ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) )
8	7	unieqi 4445	. 2 |- U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( j e. A |-> B ) ) = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) )
9		df-esum 30090	. 2 |- Σ* j e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( j e. A |-> B ) )
10		df-esum 30090	. 2 |- Σ* k e. A C = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> C ) )
11	8, 9, 10	3eqtr4i 2654	1 |- Σ* j e. A B = Σ* k e. A C

Syntax hints:   -> wi 4   = wceq 1483   F/_wnfc 2751   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-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:  cbvesumv  30105  esumfzf  30131  carsggect  30380