Theorem esumeq1 34218

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 2738	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
3	1, 2	esumeq12d 34217	1 ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1542  Σ^*cesum 34211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-iota 6458  df-fv 6510  df-ov 7373  df-esum 34212

This theorem is referenced by:  esumrnmpt  34236  esumpad  34239  esumpad2  34240  esumpr  34250  esumpr2  34251  esumfzf  34253  esumpmono  34263  esumcvg  34270  esumcvg2  34271  esum2dlem  34276  measvun  34393  ddemeas  34420  oms0  34481  omssubadd  34484  carsgsigalem  34499  carsgclctunlem1  34501  carsgclctunlem2  34503  carsgclctun  34505  pmeasmono  34508  pmeasadd  34509