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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem1 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 36434. (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 36414 | . . . . 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 36414 | . . . . 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 7443 | . . 3 ⊢ (𝜑 → ((𝑇‘𝐵) − (𝑇‘𝐴)) = ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) |
9 | 8 | fveq2d 6905 | . 2 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))) |
10 | 9 | breq1d 5159 | 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 206 = wceq 1535 ∈ wcel 2104 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6558 (class class class)co 7425 ℝcr 11145 1c1 11147 + caddc 11149 ≤ cle 11287 − cmin 11483 / cdiv 11911 2c2 12312 ⌊cfl 13816 abscabs 15259 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-iota 6510 df-fun 6560 df-fv 6566 df-ov 7428 |
This theorem is referenced by: dnibndlem2 36422 dnibndlem9 36429 dnibndlem12 36432 |
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