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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem5 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 34365. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnibndlem5 | ⊢ (𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ) | |
2 | halfre 12027 | . . . . . . 7 ⊢ (1 / 2) ∈ ℝ | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (1 / 2) ∈ ℝ) |
4 | readdcl 10795 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 + (1 / 2)) ∈ ℝ) | |
5 | 1, 3, 4 | syl2anc2 588 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + (1 / 2)) ∈ ℝ) |
6 | flltp1 13358 | . . . . 5 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (𝐴 + (1 / 2)) < ((⌊‘(𝐴 + (1 / 2))) + 1)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + (1 / 2)) < ((⌊‘(𝐴 + (1 / 2))) + 1)) |
8 | ax-1cn 10770 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
9 | 2halves 12041 | . . . . . . . . 9 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ ((1 / 2) + (1 / 2)) = 1 |
11 | 10 | eqcomi 2743 | . . . . . . 7 ⊢ 1 = ((1 / 2) + (1 / 2)) |
12 | 11 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 1 = ((1 / 2) + (1 / 2))) |
13 | 12 | oveq2d 7218 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((⌊‘(𝐴 + (1 / 2))) + 1) = ((⌊‘(𝐴 + (1 / 2))) + ((1 / 2) + (1 / 2)))) |
14 | reflcl 13354 | . . . . . . . . . 10 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
15 | 5, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
16 | 15 | recnd 10844 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℂ) |
17 | 3 | recnd 10844 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (1 / 2) ∈ ℂ) |
18 | 16, 17, 17 | 3jca 1130 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((⌊‘(𝐴 + (1 / 2))) ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (1 / 2) ∈ ℂ)) |
19 | addass 10799 | . . . . . . 7 ⊢ (((⌊‘(𝐴 + (1 / 2))) ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) + (1 / 2)) = ((⌊‘(𝐴 + (1 / 2))) + ((1 / 2) + (1 / 2)))) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) + (1 / 2)) = ((⌊‘(𝐴 + (1 / 2))) + ((1 / 2) + (1 / 2)))) |
21 | 20 | eqcomd 2740 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((⌊‘(𝐴 + (1 / 2))) + ((1 / 2) + (1 / 2))) = (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) + (1 / 2))) |
22 | 13, 21 | eqtrd 2774 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(𝐴 + (1 / 2))) + 1) = (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) + (1 / 2))) |
23 | 7, 22 | breqtrd 5069 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + (1 / 2)) < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) + (1 / 2))) |
24 | 15, 3 | jca 515 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((⌊‘(𝐴 + (1 / 2))) ∈ ℝ ∧ (1 / 2) ∈ ℝ)) |
25 | readdcl 10795 | . . . . 5 ⊢ (((⌊‘(𝐴 + (1 / 2))) ∈ ℝ ∧ (1 / 2) ∈ ℝ) → ((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) ∈ ℝ) | |
26 | 24, 25 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) ∈ ℝ) |
27 | 1, 26, 3 | ltadd1d 11408 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 < ((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) ↔ (𝐴 + (1 / 2)) < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) + (1 / 2)))) |
28 | 23, 27 | mpbird 260 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘(𝐴 + (1 / 2))) + (1 / 2))) |
29 | 1, 26 | posdifd 11402 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < ((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) ↔ 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))) |
30 | 28, 29 | mpbid 235 | 1 ⊢ (𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 < clt 10850 − cmin 11045 / cdiv 11472 2c2 11868 ⌊cfl 13348 |
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 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-n0 12074 df-z 12160 df-uz 12422 df-fl 13350 |
This theorem is referenced by: dnibndlem9 34360 |
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