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Theorem itgvallem 25284
Description: Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
itgvallem.1 (i↑𝐾) = 𝑇
Assertion
Ref Expression
itgvallem (𝑘 = 𝐾 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0))))
Distinct variable groups:   𝑥,𝑘   𝑥,𝐾
Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐵(𝑥,𝑘)   𝑇(𝑥,𝑘)   𝐾(𝑘)

Proof of Theorem itgvallem
StepHypRef Expression
1 oveq2 7412 . . . . . . . . 9 (𝑘 = 𝐾 → (i↑𝑘) = (i↑𝐾))
2 itgvallem.1 . . . . . . . . 9 (i↑𝐾) = 𝑇
31, 2eqtrdi 2789 . . . . . . . 8 (𝑘 = 𝐾 → (i↑𝑘) = 𝑇)
43oveq2d 7420 . . . . . . 7 (𝑘 = 𝐾 → (𝐵 / (i↑𝑘)) = (𝐵 / 𝑇))
54fveq2d 6892 . . . . . 6 (𝑘 = 𝐾 → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / 𝑇)))
65breq2d 5159 . . . . 5 (𝑘 = 𝐾 → (0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(𝐵 / 𝑇))))
76anbi2d 630 . . . 4 (𝑘 = 𝐾 → ((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) ↔ (𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇)))))
87, 5ifbieq1d 4551 . . 3 (𝑘 = 𝐾 → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0))
98mpteq2dv 5249 . 2 (𝑘 = 𝐾 → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0)))
109fveq2d 6892 1 (𝑘 = 𝐾 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  ifcif 4527   class class class wbr 5147  cmpt 5230  cfv 6540  (class class class)co 7404  cr 11105  0cc0 11106  ici 11108  cle 11245   / cdiv 11867  cexp 14023  cre 15040  2citg2 25115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-iota 6492  df-fv 6548  df-ov 7407
This theorem is referenced by:  iblcnlem1  25287  itgcnlem  25289
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