Theorem esumeq1 30096

Description: Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)

Assertion

Ref	Expression
esumeq1	|- ( A = B -> Σ* k e. A C = Σ* k e. B C )

Distinct variable groups:   A, k   B, k

Allowed substitution hint:   C( k)

Proof of Theorem esumeq1

Step	Hyp	Ref	Expression
1		id 22	. 2 |- ( A = B -> A = B )
2		eqidd 2623	. 2 |- ( A = B -> C = C )
3	1, 2	esumeq12d 30095	1 |- ( A = B -> Σ* k e. A C = Σ* k e. B C )

Syntax hints:   -> wi 4   = wceq 1483  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:  esumrnmpt  30114  esumpad  30117  esumpad2  30118  esumpr  30128  esumpr2  30129  esumfzf  30131  esumpmono  30141  esumcvg  30148  esumcvg2  30149  esum2dlem  30154  measvun  30272  ddemeas  30299  oms0  30359  omssubadd  30362  carsgsigalem  30377  carsgclctunlem1  30379  carsgclctunlem2  30381  carsgclctun  30383  pmeasmono  30386  pmeasadd  30387