Theorem cbvitgvw2 36476

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 7378	. . . . . . 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 2833	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
6	5	anbi1d 637	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣)))
7	6	ifbid 4478	. . . . . . 7 ⊢ (𝑥 = 𝑦 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
8	2, 7	csbeq12dv 3840	. . . . . 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 5176	. . . . 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 6830	. . . 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 7367	. . 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 36430	. 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 25608	. 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 25608	. 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 2772	1 ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⦋csb 3831  ifcif 4454   class class class wbr 5072   ↦ cmpt 5153  ‘cfv 6485  (class class class)co 7356  ℝcr 11028  0cc0 11029  ici 11031   · cmul 11034   ≤ cle 11171   / cdiv 11798  3c3 12228  ...cfz 13452  ↑cexp 14014  ℜcre 15050  Σcsu 15639  ∫₂citg2 25601  ∫citg 25603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-xp 5624  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-iota 6441  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-seq 13955  df-sum 15640  df-itg 25608

This theorem is referenced by:  cbvditgvw2  36477