Theorem itgeq12sdv 36263

Description: Equality theorem for an integral. Deduction form. General version of itgeq1d 46065 and itgeq2sdv 36264. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
itgeq12sdv.1	⊢ (𝜑 → 𝐴 = 𝐵)
itgeq12sdv.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
itgeq12sdv	⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem itgeq12sdv

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

Step	Hyp	Ref	Expression
1		itgeq12sdv.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 = 𝐷)
2	1	oveq1d 7361	. . . . . . . 8 ⊢ (𝜑 → (𝐶 / (i↑𝑘)) = (𝐷 / (i↑𝑘)))
3	2	fveq2d 6826	. . . . . . 7 ⊢ (𝜑 → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
4		itgeq12sdv.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = 𝐵)
5	4	eleq2d 2817	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
6	5	anbi1d 631	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦)))
7	6	ifbid 4496	. . . . . . 7 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
8	3, 7	csbeq12dv 3854	. . . . . 6 ⊢ (𝜑 → ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
9	8	mpteq2dv 5183	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))
10	9	fveq2d 6826	. . . 4 ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
11	10	oveq2d 7362	. . 3 ⊢ (𝜑 → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) = ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))))
12	11	sumeq2sdv 15610	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))))
13		df-itg 25551	. 2 ⊢ ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
14		df-itg 25551	. 2 ⊢ ∫𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
15	12, 13, 14	3eqtr4g 2791	1 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⦋csb 3845  ifcif 4472   class class class wbr 5089   ↦ cmpt 5170  ‘cfv 6481  (class class class)co 7346  ℝcr 11005  0cc0 11006  ici 11008   · cmul 11011   ≤ cle 11147   / cdiv 11774  3c3 12181  ...cfz 13407  ↑cexp 13968  ℜcre 15004  Σcsu 15593  ∫₂citg2 25544  ∫citg 25546

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 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-xp 5620  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-iota 6437  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seq 13909  df-sum 15594  df-itg 25551

This theorem is referenced by:  itgeq2sdv  36264  ditgeq123dv  36265