Theorem itgvallem 25754

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 7376	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = (i↑𝐾))
2		itgvallem.1	. . . . . . . . 9 ⊢ (i↑𝐾) = 𝑇
3	1, 2	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = 𝑇)
4	3	oveq2d 7384	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐵 / (i↑𝑘)) = (𝐵 / 𝑇))
5	4	fveq2d 6846	. . . . . 6 ⊢ (𝑘 = 𝐾 → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / 𝑇)))
6	5	breq2d 5112	. . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(𝐵 / 𝑇))))
7	6	anbi2d 631	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇)))))
8	7, 5	ifbieq1d 4506	. . 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 5194	. 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 6846	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 1542   ∈ wcel 2114  ifcif 4481   class class class wbr 5100   ↦ cmpt 5181  ‘cfv 6500  (class class class)co 7368  ℝcr 11037  0cc0 11038  ici 11040   ≤ cle 11179   / cdiv 11806  ↑cexp 13996  ℜcre 15032  ∫₂citg2 25585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-iota 6456  df-fv 6508  df-ov 7371

This theorem is referenced by:  iblcnlem1  25757  itgcnlem  25759