Theorem itgeq12i 36829

Description: Equality inference for an integral. General version of itgeq1i 36830 and itgeq2i 36831. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
itgeq12i.1	⊢ 𝐴 = 𝐵
itgeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
itgeq12i	⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥

Proof of Theorem itgeq12i

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

Step	Hyp	Ref	Expression
1		itgeq12i.2	. . . . . . . . . 10 ⊢ 𝐶 = 𝐷
2	1	oveq1i 7424	. . . . . . . . 9 ⊢ (𝐶 / (i↑𝑘)) = (𝐷 / (i↑𝑘))
3	2	fveq2i 6882	. . . . . . . 8 ⊢ (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
4		itgeq12i.1	. . . . . . . . . . . 12 ⊢ 𝐴 = 𝐵
5	4	eleq2i 2852	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
6	5	anbi1i 636	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦))
7		ifbi 4505	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
8	6, 7	ax-mp 5	. . . . . . . . 9 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)
9	8	ax-gen 1828	. . . . . . . 8 ⊢ ∀𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)
10	3, 9	pm3.2i 476	. . . . . . 7 ⊢ ((ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))) ∧ ∀𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
11		csbeq2 3852	. . . . . . . 8 ⊢ (∀𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0) → ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
12		csbeq1 3850	. . . . . . . 8 ⊢ ((ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))) → ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0) = ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
13	11, 12	sylan9eqr 2817	. . . . . . 7 ⊢ (((ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))) ∧ ∀𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)) → ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
14	10, 13	ax-mp 5	. . . . . 6 ⊢ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)
15	14	mpteq2i 5201	. . . . 5 ⊢ (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))
16	15	fveq2i 6882	. . . 4 ⊢ (∫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)))
17	16	oveq2i 7425	. . 3 ⊢ ((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))))
18	17	sumeq2si 36825	. 2 ⊢ 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))))
19		df-itg 25854	. 2 ⊢ ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
20		df-itg 25854	. 2 ⊢ ∫𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
21	18, 19, 20	3eqtr4i 2793	1 ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ⦋csb 3847  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6533  (class class class)co 7414  ℝcr 11126  0cc0 11127  ici 11129   · cmul 11132   ≤ cle 11271   / cdiv 11898  3c3 12323  ...cfz 13564  ↑cexp 14128  ℜcre 15187  Σcsu 15776  ∫₂citg2 25847  ∫citg 25849

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-xp 5661  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-iota 6489  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-seq 14069  df-sum 15777  df-itg 25854

This theorem is used by:  itgeq1i  36830  itgeq2i  36831  ditgeq123i  36832