Theorem itgvallem 24854

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((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 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

Step	Hyp	Ref	Expression
1		oveq2 7263	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = (i↑𝐾))
2		itgvallem.1	. . . . . . . . 9 ⊢ (i↑𝐾) = 𝑇
3	1, 2	eqtrdi 2795	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = 𝑇)
4	3	oveq2d 7271	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐵 / (i↑𝑘)) = (𝐵 / 𝑇))
5	4	fveq2d 6760	. . . . . 6 ⊢ (𝑘 = 𝐾 → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / 𝑇)))
6	5	breq2d 5082	. . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(𝐵 / 𝑇))))
7	6	anbi2d 628	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇)))))
8	7, 5	ifbieq1d 4480	. . 3 ⊢ (𝑘 = 𝐾 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0))
9	8	mpteq2dv 5172	. 2 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0)))
10	9	fveq2d 6760	1 ⊢ (𝑘 = 𝐾 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ifcif 4456   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  ℝcr 10801  0cc0 10802  ici 10804   ≤ cle 10941   / cdiv 11562  ↑cexp 13710  ℜcre 14736  ∫₂citg2 24685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  iblcnlem1  24857  itgcnlem  24859