Theorem cbvitgvw2 36209

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 7391	. . . . . . 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 2822	. . . . . . . . 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 4508	. . . . . . 7 ⊢ (𝑥 = 𝑦 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
8	2, 7	csbeq12dv 3868	. . . . . 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 5206	. . . . 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 6843	. . . 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 7380	. . 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 36163	. 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 25500	. 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 25500	. 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 2762	1 ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⦋csb 3859  ifcif 4484   class class class wbr 5102   ↦ cmpt 5183  ‘cfv 6499  (class class class)co 7369  ℝcr 11043  0cc0 11044  ici 11046   · cmul 11049   ≤ cle 11185   / cdiv 11811  3c3 12218  ...cfz 13444  ↑cexp 14002  ℜcre 15039  Σcsu 15628  ∫₂citg2 25493  ∫citg 25495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-iota 6452  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-seq 13943  df-sum 15629  df-itg 25500

This theorem is referenced by:  cbvditgvw2  36210