| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > itgvallem | Structured version Visualization version GIF version | ||
| Description: Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
| Ref | Expression |
|---|---|
| itgvallem.1 | ⊢ (i↑𝐾) = 𝑇 |
| Ref | Expression |
|---|---|
| itgvallem | ⊢ (𝑘 = 𝐾 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq2 7397 | . . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = (i↑𝐾)) | |
| 2 | itgvallem.1 | . . . . . . . . 9 ⊢ (i↑𝐾) = 𝑇 | |
| 3 | 1, 2 | eqtrdi 2781 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (i↑𝑘) = 𝑇) |
| 4 | 3 | oveq2d 7405 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝐵 / (i↑𝑘)) = (𝐵 / 𝑇)) |
| 5 | 4 | fveq2d 6864 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / 𝑇))) |
| 6 | 5 | breq2d 5121 | . . . . 5 ⊢ (𝑘 = 𝐾 → (0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) ↔ 0 ≤ (ℜ‘(𝐵 / 𝑇)))) |
| 7 | 6 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))))) |
| 8 | 7, 5 | ifbieq1d 4515 | . . 3 ⊢ (𝑘 = 𝐾 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0)) |
| 9 | 8 | mpteq2dv 5203 | . 2 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0))) |
| 10 | 9 | fveq2d 6864 | 1 ⊢ (𝑘 = 𝐾 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ifcif 4490 class class class wbr 5109 ↦ cmpt 5190 ‘cfv 6513 (class class class)co 7389 ℝcr 11073 0cc0 11074 ici 11076 ≤ cle 11215 / cdiv 11841 ↑cexp 14032 ℜcre 15069 ∫2citg2 25523 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2702 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-br 5110 df-opab 5172 df-mpt 5191 df-iota 6466 df-fv 6521 df-ov 7392 |
| This theorem is referenced by: iblcnlem1 25695 itgcnlem 25697 |
| Copyright terms: Public domain | W3C validator |