Theorem cbvitgv 25987

Description: Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Hypothesis

Ref	Expression
cbvitg.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvitgv	⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem cbvitgv

Dummy variables 𝑘 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvitg.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
2	1	fvoveq1d 7441	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
3		eleq1w 2848	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
4	3	anbi1d 643	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣)))
5	4	ifbid 4513	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
6	2, 5	csbeq12dv 3863	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
7	6	cbvmptv 5217	. . . . . . 7 ⊢ (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)) = (𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))
8	7	fveq2i 6888	. . . . . 6 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))) = (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))
9	8	oveq2i 7430	. . . . 5 ⊢ ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = ((i↑𝑘) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
10	9	a1i 11	. . . 4 ⊢ (⊤ → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = ((i↑𝑘) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))))
11	10	sumeq2sdv 15778	. . 3 ⊢ (⊤ → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))))
12	11	mptru 1577	. 2 ⊢ Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
13		df-itg 25833	. 2 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
14		df-itg 25833	. 2 ⊢ ∫𝐴𝐶 d𝑦 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑣⦌if((𝑦 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
15	12, 13, 14	3eqtr4i 2798	1 ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  ⦋csb 3854  ifcif 4489   class class class wbr 5111   ↦ cmpt 5194  ‘cfv 6540  (class class class)co 7419  ℝcr 11114  0cc0 11115  ici 11117   · cmul 11120   ≤ cle 11259   / cdiv 11886  3c3 12311  ...cfz 13551  ↑cexp 14115  ℜcre 15172  Σcsu 15761  ∫₂citg2 25826  ∫citg 25828

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 2148  ax-9 2156  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seq 14056  df-sum 15762  df-itg 25833

This theorem is used by:  ftc1a  26247  tgoldbachgtd  35114  itgiccshift  46752  itgperiod  46753  dirkeritg  46874  fourierdlem73  46951  fourierdlem82  46960  fourierdlem93  46971  fourierdlem111  46989  fourierdlem112  46990