Theorem itgvallem 25711

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 7354	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = (i↑𝐾))
2		itgvallem.1	. . . . . . . . 9 ⊢ (i↑𝐾) = 𝑇
3	1, 2	eqtrdi 2782	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = 𝑇)
4	3	oveq2d 7362	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐵 / (i↑𝑘)) = (𝐵 / 𝑇))
5	4	fveq2d 6826	. . . . . 6 ⊢ (𝑘 = 𝐾 → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / 𝑇)))
6	5	breq2d 5103	. . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(𝐵 / 𝑇))))
7	6	anbi2d 630	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇)))))
8	7, 5	ifbieq1d 4500	. . 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 5185	. 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 6826	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 1541   ∈ wcel 2111  ifcif 4475   class class class wbr 5091   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346  ℝcr 11002  0cc0 11003  ici 11005   ≤ cle 11144   / cdiv 11771  ↑cexp 13965  ℜcre 15001  ∫₂citg2 25542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-iota 6437  df-fv 6489  df-ov 7349

This theorem is referenced by:  iblcnlem1  25714  itgcnlem  25716