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Theorem itgeq1f 25820
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 2916 . . . . . 6 𝑥 𝐴 = 𝐵
4 eleq2 2827 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
54anbi1d 631 . . . . . . . . 9 (𝐴 = 𝐵 → ((𝑥𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥𝐵 ∧ 0 ≤ 𝑦)))
65ifbid 4553 . . . . . . . 8 (𝐴 = 𝐵 → if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
76csbeq2dv 3914 . . . . . . 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 5245 . . . . 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 7446 . . 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 15735 . 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 25671 . 2 𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
14 df-itg 25671 . 2 𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐶 / (i↑𝑘))) / 𝑦if((𝑥𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
1512, 13, 143eqtr4g 2799 1 (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wnfc 2887  csb 3907  ifcif 4530   class class class wbr 5147  cmpt 5230  cfv 6562  (class class class)co 7430  cr 11151  0cc0 11152  ici 11154   · cmul 11157  cle 11293   / cdiv 11917  3c3 12319  ...cfz 13543  cexp 14098  cre 15132  Σcsu 15718  2citg2 25664  citg 25666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5694  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-iota 6515  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seq 14039  df-sum 15719  df-itg 25671
This theorem is referenced by: (None)
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