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Theorem itgeq1 25808
Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
Assertion
Ref Expression
itgeq1 (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem itgeq1
Dummy variables 𝑦 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2830 . . . . . . . . 9 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 631 . . . . . . . 8 (𝐴 = 𝐵 → ((𝑥𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥𝐵 ∧ 0 ≤ 𝑦)))
32ifbid 4549 . . . . . . 7 (𝐴 = 𝐵 → if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
43csbeq2dv 3906 . . . . . 6 (𝐴 = 𝐵(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
54mpteq2dv 5244 . . . . 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)))
65fveq2d 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))))
76oveq2d 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)))))
87sumeq2sdv 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)))))
9 df-itg 25658 . 2 𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
10 df-itg 25658 . 2 𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
118, 9, 103eqtr4g 2802 1 (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  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-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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  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:  itgsplitioo  25873  ditgeq1  25883  ditgeq2  25884  ditg0  25888  ditgneg  25892  ftc1lem1  26076  ftc1a  26078  ftc2  26085  itgsubstlem  26089  ftc1anc  37708  ftc2nc  37709  areacirc  37720  itgeq1d  45972  fourierdlem103  46224  fourierdlem104  46225
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