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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem1 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 34716. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnibndlem1.1 | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
dnibndlem1.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dnibndlem1.3 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
dnibndlem1 | ⊢ (𝜑 → ((abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnibndlem1.3 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | dnibndlem1.1 | . . . . . 6 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
3 | 2 | dnival 34696 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝑇‘𝐵) = (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) |
4 | 1, 3 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑇‘𝐵) = (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) |
5 | dnibndlem1.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
6 | 2 | dnival 34696 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
8 | 4, 7 | oveq12d 7325 | . . 3 ⊢ (𝜑 → ((𝑇‘𝐵) − (𝑇‘𝐴)) = ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) |
9 | 8 | fveq2d 6808 | . 2 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))) |
10 | 9 | breq1d 5091 | 1 ⊢ (𝜑 → ((abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2104 class class class wbr 5081 ↦ cmpt 5164 ‘cfv 6458 (class class class)co 7307 ℝcr 10916 1c1 10918 + caddc 10920 ≤ cle 11056 − cmin 11251 / cdiv 11678 2c2 12074 ⌊cfl 13556 abscabs 14990 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-iota 6410 df-fun 6460 df-fv 6466 df-ov 7310 |
This theorem is referenced by: dnibndlem2 34704 dnibndlem9 34711 dnibndlem12 34714 |
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