Theorem esumeq1 34017

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

Assertion

Ref	Expression
esumeq1	⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

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 2730	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
3	1, 2	esumeq12d 34016	1 ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1540  Σ^*cesum 34010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-iota 6452  df-fv 6507  df-ov 7372  df-esum 34011

This theorem is referenced by:  esumrnmpt  34035  esumpad  34038  esumpad2  34039  esumpr  34049  esumpr2  34050  esumfzf  34052  esumpmono  34062  esumcvg  34069  esumcvg2  34070  esum2dlem  34075  measvun  34192  ddemeas  34219  oms0  34281  omssubadd  34284  carsgsigalem  34299  carsgclctunlem1  34301  carsgclctunlem2  34303  carsgclctun  34305  pmeasmono  34308  pmeasadd  34309