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Theorem itgeq1f 25806
Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) Avoid axioms. (Revised by GG, 1-Sep-2025.)
Hypotheses
Ref Expression
itgeq1f.1 𝑥𝐴
itgeq1f.2 𝑥𝐵
Assertion
Ref Expression
itgeq1f (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Proof of Theorem itgeq1f
Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itgeq1f.1 . . . . . . 7 𝑥𝐴
2 itgeq1f.2 . . . . . . 7 𝑥𝐵
31, 2nfeq 2919 . . . . . 6 𝑥 𝐴 = 𝐵
4 eleq2 2830 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 631 . . . . . . . . 9 (𝐴 = 𝐵 → ((𝑥𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥𝐵 ∧ 0 ≤ 𝑦)))
65ifbid 4549 . . . . . . . 8 (𝐴 = 𝐵 → if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
76csbeq2dv 3906 . . . . . . 7 (𝐴 = 𝐵(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
87adantr 480 . . . . . 6 ((𝐴 = 𝐵𝑥 ∈ ℝ) → (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
93, 8mpteq2da 5240 . . . . 5 (𝐴 = 𝐵 → (𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))
109fveq2d 6910 . . . 4 (𝐴 = 𝐵 → (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
1110oveq2d 7447 . . 3 (𝐴 = 𝐵 → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) = ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))))
1211sumeq2sdv 15739 . 2 (𝐴 = 𝐵 → 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‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))))
13 df-itg 25658 . 2 𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
14 df-itg 25658 . 2 𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
1512, 13, 143eqtr4g 2802 1 (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wnfc 2890  csb 3899  ifcif 4525   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155  ici 11157   · cmul 11160  cle 11296   / cdiv 11920  3c3 12322  ...cfz 13547  cexp 14102  cre 15136  Σcsu 15722  2citg2 25651  citg 25653
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
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-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seq 14043  df-sum 15723  df-itg 25658
This theorem is referenced by: (None)
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