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Theorem cbvitgvw2 36216
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 . . . . . . . 8 (𝑥 = 𝑦𝐶 = 𝐷)
21fvoveq1d 7472 . . . . . . 7 (𝑥 = 𝑦 → (ℜ‘(𝐶 / (i↑𝑡))) = (ℜ‘(𝐷 / (i↑𝑡))))
3 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
4 cbvitgvw2.2 . . . . . . . . . 10 (𝑥 = 𝑦𝐴 = 𝐵)
53, 4eleq12d 2838 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
65anbi1d 630 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦𝐵 ∧ 0 ≤ 𝑣)))
76ifbid 4571 . . . . . . 7 (𝑥 = 𝑦 → if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
82, 7csbeq12dv 3930 . . . . . 6 (𝑥 = 𝑦(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
98cbvmptv 5279 . . . . 5 (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 6925 . . . 4 (∫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 7461 . . 3 ((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))))
1211sumeq2si 36168 . 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))))
13 df-itg 25679 . 2 𝐴𝐶 d𝑥 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑡))) / 𝑣if((𝑥𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
14 df-itg 25679 . 2 𝐵𝐷 d𝑦 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑡))) / 𝑣if((𝑦𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))))
1512, 13, 143eqtr4i 2778 1 𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  csb 3921  ifcif 4548   class class class wbr 5166  cmpt 5249  cfv 6575  (class class class)co 7450  cr 11185  0cc0 11186  ici 11188   · cmul 11191  cle 11327   / cdiv 11949  3c3 12351  ...cfz 13569  cexp 14114  cre 15148  Σcsu 15736  2citg2 25672  citg 25674
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 2110  ax-9 2118  ax-ext 2711
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 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-iota 6527  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-seq 14055  df-sum 15737  df-itg 25679
This theorem is referenced by:  cbvditgvw2  36217
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