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Theorem itgeq12sdv 36162
Description: Equality theorem for an integral. Deduction form. General version of itgeq1d 45863 and itgeq2sdv 36163. (Contributed by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
itgeq12sdv.1 (𝜑𝐴 = 𝐵)
itgeq12sdv.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
itgeq12sdv (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem itgeq12sdv
Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itgeq12sdv.2 . . . . . . . . 9 (𝜑𝐶 = 𝐷)
21oveq1d 7440 . . . . . . . 8 (𝜑 → (𝐶 / (i↑𝑘)) = (𝐷 / (i↑𝑘)))
32fveq2d 6905 . . . . . . 7 (𝜑 → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
4 itgeq12sdv.1 . . . . . . . . . 10 (𝜑𝐴 = 𝐵)
54eleq2d 2823 . . . . . . . . 9 (𝜑 → (𝑥𝐴𝑥𝐵))
65anbi1d 630 . . . . . . . 8 (𝜑 → ((𝑥𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥𝐵 ∧ 0 ≤ 𝑦)))
76ifbid 4553 . . . . . . 7 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
83, 7csbeq12dv 3917 . . . . . 6 (𝜑(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
98mpteq2dv 5251 . . . . 5 (𝜑 → (𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))
109fveq2d 6905 . . . 4 (𝜑 → (∫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))))
1110oveq2d 7441 . . 3 (𝜑 → ((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)))))
1211sumeq2sdv 15725 . 2 (𝜑 → 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)))))
13 df-itg 25653 . 2 𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
14 df-itg 25653 . 2 𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐷 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
1512, 13, 143eqtr4g 2798 1 (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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:  itgeq2sdv  36163  ditgeq123dv  36164
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