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Theorem itgeq12i 36185
Description: Equality inference for an integral. General version of itgeq1i 36186 and itgeq2i 36187. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
itgeq12i.1 𝐴 = 𝐵
itgeq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
itgeq12i 𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥

Proof of Theorem itgeq12i
Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itgeq12i.2 . . . . . . . . . 10 𝐶 = 𝐷
21oveq1i 7439 . . . . . . . . 9 (𝐶 / (i↑𝑘)) = (𝐷 / (i↑𝑘))
32fveq2i 6907 . . . . . . . 8 (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
4 itgeq12i.1 . . . . . . . . . . . 12 𝐴 = 𝐵
54eleq2i 2832 . . . . . . . . . . 11 (𝑥𝐴𝑥𝐵)
65anbi1i 624 . . . . . . . . . 10 ((𝑥𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥𝐵 ∧ 0 ≤ 𝑦))
7 ifbi 4546 . . . . . . . . . 10 (((𝑥𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥𝐵 ∧ 0 ≤ 𝑦)) → if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
86, 7ax-mp 5 . . . . . . . . 9 if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)
98ax-gen 1795 . . . . . . . 8 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)
103, 9pm3.2i 470 . . . . . . 7 ((ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))) ∧ ∀𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
11 csbeq2 3903 . . . . . . . 8 (∀𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0) → (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
12 csbeq1 3901 . . . . . . . 8 ((ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))) → (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0) = (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
1311, 12sylan9eqr 2798 . . . . . . 7 (((ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))) ∧ ∀𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)) → (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
1410, 13ax-mp 5 . . . . . 6 (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)
1514mpteq2i 5245 . . . . 5 (𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
1615fveq2i 6907 . . . 4 (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))
1716oveq2i 7440 . . 3 ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) = ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
1817sumeq2si 36181 . 2 Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
19 df-itg 25648 . 2 𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
20 df-itg 25648 . 2 𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
2118, 19, 203eqtr4i 2774 1 𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  csb 3898  ifcif 4524   class class class wbr 5141  cmpt 5223  cfv 6559  (class class class)co 7429  cr 11150  0cc0 11151  ici 11153   · cmul 11156  cle 11292   / cdiv 11916  3c3 12318  ...cfz 13543  cexp 14098  cre 15132  Σcsu 15718  2citg2 25641  citg 25643
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 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-iota 6512  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-seq 14039  df-sum 15719  df-itg 25648
This theorem is referenced by:  itgeq1i  36186  itgeq2i  36187  ditgeq123i  36188
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