Theorem esumeq1 33032

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 2734	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
3	1, 2	esumeq12d 33031	1 ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1542  Σ^*cesum 33025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-iota 6496  df-fv 6552  df-ov 7412  df-esum 33026

This theorem is referenced by:  esumrnmpt  33050  esumpad  33053  esumpad2  33054  esumpr  33064  esumpr2  33065  esumfzf  33067  esumpmono  33077  esumcvg  33084  esumcvg2  33085  esum2dlem  33090  measvun  33207  ddemeas  33234  oms0  33296  omssubadd  33299  carsgsigalem  33314  carsgclctunlem1  33316  carsgclctunlem2  33318  carsgclctun  33320  pmeasmono  33323  pmeasadd  33324