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Theorem cbvitgdavw 36224
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 7447 . . . . . . 7 ((𝜑𝑥 = 𝑦) → (ℜ‘(𝐵 / (i↑𝑡))) = (ℜ‘(𝐶 / (i↑𝑡))))
3 eleq1w 2820 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
43adantl 481 . . . . . . . . 9 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐴))
54anbi1d 630 . . . . . . . 8 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦𝐴 ∧ 0 ≤ 𝑣)))
65ifbid 4553 . . . . . . 7 ((𝜑𝑥 = 𝑦) → if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
72, 6csbeq12dv 3917 . . . . . 6 ((𝜑𝑥 = 𝑦) → (ℜ‘(𝐵 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
87cbvmptdavw 36210 . . . . 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 6905 . . . 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 7441 . . 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 15725 . 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 25653 . 2 𝐴𝐵 d𝑥 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
13 df-itg 25653 . 2 𝐴𝐶 d𝑦 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑦𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
1411, 12, 133eqtr4g 2798 1 (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104  csb 3908  ifcif 4530   class class class wbr 5149  cmpt 5232  cfv 6558  (class class class)co 7425  cr 11145  0cc0 11146  ici 11148   · cmul 11151  cle 11287   / cdiv 11911  3c3 12313  ...cfz 13537  cexp 14088  cre 15122  Σcsu 15708  2citg2 25646  citg 25648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6317  df-iota 6510  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-seq 14029  df-sum 15709  df-itg 25653
This theorem is referenced by:  cbvditgdavw  36225
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