Theorem cbvitgdavw2 36331

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

Hypotheses

Ref	Expression
cbvitgdavw2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
cbvitgdavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvitgdavw2	⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦)

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

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

Proof of Theorem cbvitgdavw2

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

Step	Hyp	Ref	Expression
1		cbvitgdavw2.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
2	1	fvoveq1d 7363	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (ℜ‘(𝐶 / (i↑𝑡))) = (ℜ‘(𝐷 / (i↑𝑡))))
3		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
4		cbvitgdavw2.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
5	3, 4	eleq12d 2825	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
6	5	anbi1d 631	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣)))
7	6	ifbid 4494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))
8	2, 7	csbeq12dv 3854	. . . . . 6 ⊢ ((𝜑 ∧ 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	cbvmptdavw 36301	. . . . 5 ⊢ (𝜑 → (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	fveq2d 6821	. . . 4 ⊢ (𝜑 → (∫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	oveq2d 7357	. . 3 ⊢ (𝜑 → ((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	sumeq2sdv 15605	. 2 ⊢ (𝜑 → 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 25546	. 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 25546	. 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	3eqtr4g 2791	1 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⦋csb 3845  ifcif 4470   class class class wbr 5086   ↦ cmpt 5167  ‘cfv 6476  (class class class)co 7341  ℝcr 11000  0cc0 11001  ici 11003   · cmul 11006   ≤ cle 11142   / cdiv 11769  3c3 12176  ...cfz 13402  ↑cexp 13963  ℜcre 14999  Σcsu 15588  ∫₂citg2 25539  ∫citg 25541

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-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5617  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-iota 6432  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-seq 13904  df-sum 15589  df-itg 25546

This theorem is referenced by:  cbvditgdavw2  36332