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| Description: Lemma for dnibnd 36492. (Contributed by Asger C. Ipsen, 4-Apr-2021.) | 
| Ref | Expression | 
|---|---|
| dnibndlem12.1 | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | 
| dnibndlem12.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| dnibndlem12.3 | ⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| dnibndlem12.4 | ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) | 
| Ref | Expression | 
|---|---|
| dnibndlem12 | ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dnibndlem12.3 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 2 | 1 | dnicld1 36473 | . . . . . 6 ⊢ (𝜑 → (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) ∈ ℝ) | 
| 3 | dnibndlem12.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 4 | 3 | dnicld1 36473 | . . . . . 6 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) | 
| 5 | 2, 4 | resubcld 11691 | . . . . 5 ⊢ (𝜑 → ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ∈ ℝ) | 
| 6 | 5 | recnd 11289 | . . . 4 ⊢ (𝜑 → ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ∈ ℂ) | 
| 7 | 6 | abscld 15475 | . . 3 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ∈ ℝ) | 
| 8 | 1red 11262 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 9 | 1, 3 | resubcld 11691 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) | 
| 10 | 9 | recnd 11289 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) | 
| 11 | 10 | abscld 15475 | . . 3 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℝ) | 
| 12 | 8 | rehalfcld 12513 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℝ) | 
| 13 | 3, 1 | dnibndlem11 36489 | . . . 4 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (1 / 2)) | 
| 14 | halflt1 12484 | . . . . . 6 ⊢ (1 / 2) < 1 | |
| 15 | halfre 12480 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
| 16 | 1re 11261 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
| 17 | 15, 16 | pm3.2i 470 | . . . . . . 7 ⊢ ((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) | 
| 18 | ltle 11349 | . . . . . . 7 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2) < 1 → (1 / 2) ≤ 1)) | |
| 19 | 17, 18 | ax-mp 5 | . . . . . 6 ⊢ ((1 / 2) < 1 → (1 / 2) ≤ 1) | 
| 20 | 14, 19 | ax-mp 5 | . . . . 5 ⊢ (1 / 2) ≤ 1 | 
| 21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 / 2) ≤ 1) | 
| 22 | 7, 12, 8, 13, 21 | letrd 11418 | . . 3 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 1) | 
| 23 | dnibndlem12.4 | . . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) | |
| 24 | 3, 1, 23 | dnibndlem10 36488 | . . . 4 ⊢ (𝜑 → 1 ≤ (𝐵 − 𝐴)) | 
| 25 | 9 | leabsd 15453 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (abs‘(𝐵 − 𝐴))) | 
| 26 | 8, 9, 11, 24, 25 | letrd 11418 | . . 3 ⊢ (𝜑 → 1 ≤ (abs‘(𝐵 − 𝐴))) | 
| 27 | 7, 8, 11, 22, 26 | letrd 11418 | . 2 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (abs‘(𝐵 − 𝐴))) | 
| 28 | dnibndlem12.1 | . . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
| 29 | 28, 3, 1 | dnibndlem1 36479 | . 2 ⊢ (𝜑 → ((abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)) ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (abs‘(𝐵 − 𝐴)))) | 
| 30 | 27, 29 | mpbird 257 | 1 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 ⌊cfl 13830 abscabs 15273 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fl 13832 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 | 
| This theorem is referenced by: dnibndlem13 36491 | 
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