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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem1 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 34408. (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 34388 | . . . . 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 34388 | . . . . 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 7231 | . . 3 ⊢ (𝜑 → ((𝑇‘𝐵) − (𝑇‘𝐴)) = ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) |
9 | 8 | fveq2d 6721 | . 2 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))) |
10 | 9 | breq1d 5063 | 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 209 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 1c1 10730 + caddc 10732 ≤ cle 10868 − cmin 11062 / cdiv 11489 2c2 11885 ⌊cfl 13365 abscabs 14797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-ov 7216 |
This theorem is referenced by: dnibndlem2 34396 dnibndlem9 34403 dnibndlem12 34406 |
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