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Theorem cbvitgdavw 36260
Description: Change bound variable in an integral. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbvitgdavw.1 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
Assertion
Ref Expression
cbvitgdavw (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦)
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvitgdavw
Dummy variables 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvitgdavw.1 . . . . . . . 8 ((𝜑𝑥 = 𝑦) → 𝐵 = 𝐶)
21fvoveq1d 7451 . . . . . . 7 ((𝜑𝑥 = 𝑦) → (ℜ‘(𝐵 / (i↑𝑡))) = (ℜ‘(𝐶 / (i↑𝑡))))
3 eleq1w 2823 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43adantl 481 . . . . . . . . 9 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐴))
54anbi1d 631 . . . . . . . 8 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦𝐴 ∧ 0 ≤ 𝑣)))
65ifbid 4547 . . . . . . 7 ((𝜑𝑥 = 𝑦) → if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
72, 6csbeq12dv 3907 . . . . . 6 ((𝜑𝑥 = 𝑦) → (ℜ‘(𝐵 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
87cbvmptdavw 36246 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)) = (𝑦 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))
98fveq2d 6908 . . . 4 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))) = (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
109oveq2d 7445 . . 3 (𝜑 → ((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = ((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))))
1110sumeq2sdv 15735 . 2 (𝜑 → Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))))
12 df-itg 25648 . 2 𝐴𝐵 d𝑥 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
13 df-itg 25648 . 2 𝐴𝐶 d𝑦 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
1411, 12, 133eqtr4g 2801 1 (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = 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:  cbvditgdavw  36261
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