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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem1 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 33064. (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 33044 | . . . . 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 33044 | . . . . 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 6940 | . . 3 ⊢ (𝜑 → ((𝑇‘𝐵) − (𝑇‘𝐴)) = ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) |
9 | 8 | fveq2d 6450 | . 2 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))) |
10 | 9 | breq1d 4896 | 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 198 = wceq 1601 ∈ wcel 2107 class class class wbr 4886 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 1c1 10273 + caddc 10275 ≤ cle 10412 − cmin 10606 / cdiv 11032 2c2 11430 ⌊cfl 12910 abscabs 14381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 |
This theorem is referenced by: dnibndlem2 33052 dnibndlem9 33059 dnibndlem12 33062 |
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