Theorem itgvallem 26085

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 7420	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = (i↑𝐾))
2		itgvallem.1	. . . . . . . . 9 ⊢ (i↑𝐾) = 𝑇
3	1, 2	eqtrdi 2812	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = 𝑇)
4	3	oveq2d 7428	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐵 / (i↑𝑘)) = (𝐵 / 𝑇))
5	4	fveq2d 6881	. . . . . 6 ⊢ (𝑘 = 𝐾 → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / 𝑇)))
6	5	breq2d 5115	. . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(𝐵 / 𝑇))))
7	6	anbi2d 642	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇)))))
8	7, 5	ifbieq1d 4507	. . 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 5199	. 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 6881	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))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6531  (class class class)co 7412  ℝcr 11180  0cc0 11181  ici 11183   ≤ cle 11325   / cdiv 11954  ↑cexp 14184  ℜcre 15244  ∫₂citg2 25917

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-iota 6487  df-fv 6539  df-ov 7415

This theorem is used by:  iblcnlem1  26088  itgcnlem  26090