Theorem itgeq1 25711

Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)

Assertion

Ref	Expression
itgeq1	⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem itgeq1

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

Step	Hyp	Ref	Expression
1		eleq2 2822	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	anbi1d 631	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦)))
3	2	ifbid 4500	. . . . . . 7 ⊢ (𝐴 = 𝐵 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
4	3	csbeq2dv 3854	. . . . . 6 ⊢ (𝐴 = 𝐵 → ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))
5	4	mpteq2dv 5189	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))
6	5	fveq2d 6835	. . . 4 ⊢ (𝐴 = 𝐵 → (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
7	6	oveq2d 7371	. . 3 ⊢ (𝐴 = 𝐵 → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) = ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))))
8	7	sumeq2sdv 15620	. 2 ⊢ (𝐴 = 𝐵 → 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‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)))))
9		df-itg 25561	. 2 ⊢ ∫𝐴𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
10		df-itg 25561	. 2 ⊢ ∫𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0))))
11	8, 9, 10	3eqtr4g 2793	1 ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⦋csb 3847  ifcif 4476   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6489  (class class class)co 7355  ℝcr 11015  0cc0 11016  ici 11018   · cmul 11021   ≤ cle 11157   / cdiv 11784  3c3 12191  ...cfz 13417  ↑cexp 13978  ℜcre 15014  Σcsu 15603  ∫₂citg2 25554  ∫citg 25556

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 2115  ax-9 2123  ax-ext 2705

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 2712  df-cleq 2725  df-clel 2808  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-xp 5627  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-iota 6445  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-seq 13919  df-sum 15604  df-itg 25561

This theorem is referenced by:  itgsplitioo  25776  ditgeq1  25786  ditgeq2  25787  ditg0  25791  ditgneg  25795  ftc1lem1  25979  ftc1a  25981  ftc2  25988  itgsubstlem  25992  ftc1anc  37751  ftc2nc  37752  areacirc  37763  itgeq1d  46069  fourierdlem103  46321  fourierdlem104  46322