Theorem itgeq12i 36566

Description: Equality inference for an integral. General version of itgeq1i 36567 and itgeq2i 36568. (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 7406	. . . . . . . . 9 ⊢ (𝐶 / (i↑𝑘)) = (𝐷 / (i↑𝑘))
3	2	fveq2i 6870	. . . . . . . 8 ⊢ (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
4		itgeq12i.1	. . . . . . . . . . . 12 ⊢ 𝐴 = 𝐵
5	4	eleq2i 2854	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
6	5	anbi1i 633	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦))
7		ifbi 4503	. . . . . . . . . 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 1815	. . . . . . . 8 ⊢ ∀𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)
10	3, 9	pm3.2i 474	. . . . . . 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 3857	. . . . . . . 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 3855	. . . . . . . 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 2819	. . . . . . 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 5196	. . . . 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 6870	. . . 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 7407	. . 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 36562	. 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 25685	. 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 25685	. 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 2795	1 ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥

Syntax hints:   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560   ∈ wcel 2142  ⦋csb 3852  ifcif 4480   class class class wbr 5100   ↦ cmpt 5181  ‘cfv 6521  (class class class)co 7396  ℝcr 11072  0cc0 11073  ici 11075   · cmul 11078   ≤ cle 11217   / cdiv 11844  3c3 12273  ...cfz 13512  ↑cexp 14074  ℜcre 15124  Σcsu 15713  ∫₂citg2 25678  ∫citg 25680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5653  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-iota 6477  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-seq 14015  df-sum 15714  df-itg 25685

This theorem is referenced by:  itgeq1i  36567  itgeq2i  36568  ditgeq123i  36569