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Mirrors > Home > MPE Home > Th. List > itgeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
itgeq1 | ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2828 | . . . . . . . . 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 4554 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ 𝑦), 𝑦, 0)) |
4 | 3 | csbeq2dv 3915 | . . . . . 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 5250 | . . . . 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 6911 | . . . 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 7447 | . . 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 15736 | . 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 25672 | . 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 25672 | . 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 2800 | 1 ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⦋csb 3908 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 ici 11155 · cmul 11158 ≤ cle 11294 / cdiv 11918 3c3 12320 ...cfz 13544 ↑cexp 14099 ℜcre 15133 Σcsu 15719 ∫2citg2 25665 ∫citg 25667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seq 14040 df-sum 15720 df-itg 25672 |
This theorem is referenced by: itgsplitioo 25888 ditgeq1 25898 ditgeq2 25899 ditg0 25903 ditgneg 25907 ftc1lem1 26091 ftc1a 26093 ftc2 26100 itgsubstlem 26104 ftc1anc 37688 ftc2nc 37689 areacirc 37700 itgeq1d 45913 fourierdlem103 46165 fourierdlem104 46166 |
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