Theorem cbvitgvw2 36208

Description: Change bound variable and domain in an integral, using implicit substitution. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvitgvw2.1	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
cbvitgvw2.2	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvitgvw2	⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦

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

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

Proof of Theorem cbvitgvw2

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

Step	Hyp	Ref	Expression
1		cbvitgvw2.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
2	1	fvoveq1d 7435	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (ℜ‘(𝐶 / (i↑𝑡))) = (ℜ‘(𝐷 / (i↑𝑡))))
3		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4		cbvitgvw2.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
5	3, 4	eleq12d 2827	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
6	5	anbi1d 631	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣)))
7	6	ifbid 4529	. . . . . . 7 ⊢ (𝑥 = 𝑦 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
8	2, 7	csbeq12dv 3888	. . . . . 6 ⊢ (𝑥 = 𝑦 → ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
9	8	cbvmptv 5235	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)) = (𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
10	9	fveq2i 6889	. . . 4 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))) = (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)))
11	10	oveq2i 7424	. . 3 ⊢ ((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = ((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))))
12	11	sumeq2si 36162	. 2 ⊢ Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))))
13		df-itg 25594	. 2 ⊢ ∫𝐴𝐶 d𝑥 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))))
14		df-itg 25594	. 2 ⊢ ∫𝐵𝐷 d𝑦 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))))
15	12, 13, 14	3eqtr4i 2767	1 ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ⦋csb 3879  ifcif 4505   class class class wbr 5123   ↦ cmpt 5205  ‘cfv 6541  (class class class)co 7413  ℝcr 11136  0cc0 11137  ici 11139   · cmul 11142   ≤ cle 11278   / cdiv 11902  3c3 12304  ...cfz 13529  ↑cexp 14084  ℜcre 15118  Σcsu 15704  ∫₂citg2 25587  ∫citg 25589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-xp 5671  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-iota 6494  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-seq 14025  df-sum 15705  df-itg 25594

This theorem is referenced by:  cbvditgvw2  36209