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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvitgvw2 | Structured version Visualization version GIF version |
Description: Change bound variable and domain in an integral, using implicit substitution. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvitgvw2.1 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
cbvitgvw2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvitgvw2 | ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvitgvw2.1 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
2 | 1 | fvoveq1d 7472 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (ℜ‘(𝐶 / (i↑𝑡))) = (ℜ‘(𝐷 / (i↑𝑡)))) |
3 | id 22 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
4 | cbvitgvw2.2 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
5 | 3, 4 | eleq12d 2838 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
6 | 5 | anbi1d 630 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣) ↔ (𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣))) |
7 | 6 | ifbid 4571 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)) |
8 | 2, 7 | csbeq12dv 3930 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0) = ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)) |
9 | 8 | cbvmptv 5279 | . . . . 5 ⊢ (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)) = (𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)) |
10 | 9 | fveq2i 6925 | . . . 4 ⊢ (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0))) = (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0))) |
11 | 10 | oveq2i 7461 | . . 3 ⊢ ((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = ((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)))) |
12 | 11 | sumeq2si 36168 | . 2 ⊢ Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)))) |
13 | df-itg 25679 | . 2 ⊢ ∫𝐴𝐶 d𝑥 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐶 / (i↑𝑡))) / 𝑣⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑣), 𝑣, 0)))) | |
14 | df-itg 25679 | . 2 ⊢ ∫𝐵𝐷 d𝑦 = Σ𝑡 ∈ (0...3)((i↑𝑡) · (∫2‘(𝑦 ∈ ℝ ↦ ⦋(ℜ‘(𝐷 / (i↑𝑡))) / 𝑣⦌if((𝑦 ∈ 𝐵 ∧ 0 ≤ 𝑣), 𝑣, 0)))) | |
15 | 12, 13, 14 | 3eqtr4i 2778 | 1 ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ⦋csb 3921 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6575 (class class class)co 7450 ℝcr 11185 0cc0 11186 ici 11188 · cmul 11191 ≤ cle 11327 / cdiv 11949 3c3 12351 ...cfz 13569 ↑cexp 14114 ℜcre 15148 Σcsu 15736 ∫2citg2 25672 ∫citg 25674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-xp 5706 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-iota 6527 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-seq 14055 df-sum 15737 df-itg 25679 |
This theorem is referenced by: cbvditgvw2 36217 |
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