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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem8 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 33830. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnibndlem8.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
dnibndlem8 | ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnibndlem8.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | halfre 11852 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
4 | 1, 3 | jca 514 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ)) |
5 | simpl 485 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → 𝐴 ∈ ℝ) | |
6 | 2 | a1i 11 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (1 / 2) ∈ ℝ) |
7 | 5, 6 | readdcld 10670 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 + (1 / 2)) ∈ ℝ) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
9 | reflcl 13167 | . . . . 5 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
11 | 1, 10 | resubcld 11068 | . . 3 ⊢ (𝜑 → (𝐴 − (⌊‘(𝐴 + (1 / 2)))) ∈ ℝ) |
12 | 1 | dnicld1 33811 | . . 3 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) |
13 | 11 | leabsd 14774 | . . . 4 ⊢ (𝜑 → (𝐴 − (⌊‘(𝐴 + (1 / 2)))) ≤ (abs‘(𝐴 − (⌊‘(𝐴 + (1 / 2)))))) |
14 | 1 | recnd 10669 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
15 | 10 | recnd 10669 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℂ) |
16 | 14, 15 | abssubd 14813 | . . . 4 ⊢ (𝜑 → (abs‘(𝐴 − (⌊‘(𝐴 + (1 / 2))))) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
17 | 13, 16 | breqtrd 5092 | . . 3 ⊢ (𝜑 → (𝐴 − (⌊‘(𝐴 + (1 / 2)))) ≤ (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
18 | 11, 12, 3, 17 | lesub2dd 11257 | . 2 ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ ((1 / 2) − (𝐴 − (⌊‘(𝐴 + (1 / 2)))))) |
19 | 3 | recnd 10669 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
20 | 19, 14, 15 | subsub3d 11027 | . . 3 ⊢ (𝜑 → ((1 / 2) − (𝐴 − (⌊‘(𝐴 + (1 / 2))))) = (((1 / 2) + (⌊‘(𝐴 + (1 / 2)))) − 𝐴)) |
21 | 19, 15 | addcomd 10842 | . . . 4 ⊢ (𝜑 → ((1 / 2) + (⌊‘(𝐴 + (1 / 2)))) = ((⌊‘(𝐴 + (1 / 2))) + (1 / 2))) |
22 | 21 | oveq1d 7171 | . . 3 ⊢ (𝜑 → (((1 / 2) + (⌊‘(𝐴 + (1 / 2)))) − 𝐴) = (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
23 | 20, 22 | eqtrd 2856 | . 2 ⊢ (𝜑 → ((1 / 2) − (𝐴 − (⌊‘(𝐴 + (1 / 2))))) = (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
24 | 18, 23 | breqtrd 5092 | 1 ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 1c1 10538 + caddc 10540 ≤ cle 10676 − cmin 10870 / cdiv 11297 2c2 11693 ⌊cfl 13161 abscabs 14593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fl 13163 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 |
This theorem is referenced by: dnibndlem9 33825 |
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