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Theorem cbvitgvw2 36163
Description: Change bound variable and domain in an integral, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvitgvw2.1 (𝑥 = 𝑦𝐶 = 𝐷)
cbvitgvw2.2 (𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
cbvitgvw2 𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvitgvw2
Dummy variables 𝑡 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvitgvw2.1 . . . . . . . . . 10 (𝑥 = 𝑦𝐶 = 𝐷)
21fvoveq1d 7467 . . . . . . . . 9 (𝑥 = 𝑦 → (ℜ‘(𝐶 / (i↑𝑡))) = (ℜ‘(𝐷 / (i↑𝑡))))
3 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑦𝑥 = 𝑦)
4 cbvitgvw2.2 . . . . . . . . . . . 12 (𝑥 = 𝑦𝐴 = 𝐵)
53, 4eleq12d 2832 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
65anbi1d 630 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝑥𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦𝐵 ∧ 0 ≤ 𝑣)))
76ifbid 4571 . . . . . . . . 9 (𝑥 = 𝑦 → if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
82, 7csbeq12dv 3924 . . . . . . . 8 (𝑥 = 𝑦(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
98cbvmptv 5282 . . . . . . 7 (𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)) = (𝑦 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
109fveq2i 6922 . . . . . 6 (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))) = (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)))
1110oveq2i 7456 . . . . 5 ((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = ((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))))
1211a1i 11 . . . 4 (⊤ → ((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = ((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)))))
1312sumeq2sdv 15747 . . 3 (⊤ → Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)))))
1413mptru 1544 . 2 Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))))
15 df-itg 25670 . 2 𝐴𝐶 d𝑥 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
16 df-itg 25670 . 2 𝐵𝐷 d𝑦 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))))
1714, 15, 163eqtr4i 2772 1 𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wtru 1538  wcel 2103  csb 3915  ifcif 4548   class class class wbr 5169  cmpt 5252  cfv 6572  (class class class)co 7445  cr 11179  0cc0 11180  ici 11182   · cmul 11185  cle 11321   / cdiv 11943  3c3 12345  ...cfz 13563  cexp 14108  cre 15142  Σcsu 15730  2citg2 25663  citg 25665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-xp 5705  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-iota 6524  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-seq 14049  df-sum 15731  df-itg 25670
This theorem is referenced by: (None)
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