Theorem esumeq1 34031

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 2731	. 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐶)
3	1, 2	esumeq12d 34030	1 ⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)

Syntax hints:   → wi 4   = wceq 1540  Σ^*cesum 34024

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-iota 6467  df-fv 6522  df-ov 7393  df-esum 34025

This theorem is referenced by:  esumrnmpt  34049  esumpad  34052  esumpad2  34053  esumpr  34063  esumpr2  34064  esumfzf  34066  esumpmono  34076  esumcvg  34083  esumcvg2  34084  esum2dlem  34089  measvun  34206  ddemeas  34233  oms0  34295  omssubadd  34298  carsgsigalem  34313  carsgclctunlem1  34315  carsgclctunlem2  34317  carsgclctun  34319  pmeasmono  34322  pmeasadd  34323