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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem1 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 33825. (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 33805 | . . . . 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 33805 | . . . . 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 7168 | . . 3 ⊢ (𝜑 → ((𝑇‘𝐵) − (𝑇‘𝐴)) = ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) |
9 | 8 | fveq2d 6668 | . 2 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))) |
10 | 9 | breq1d 5068 | 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 208 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 1c1 10532 + caddc 10534 ≤ cle 10670 − cmin 10864 / cdiv 11291 2c2 11686 ⌊cfl 13154 abscabs 14587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 |
This theorem is referenced by: dnibndlem2 33813 dnibndlem9 33820 dnibndlem12 33823 |
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