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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iooabslt | Structured version Visualization version GIF version |
Description: An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iooabslt.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
iooabslt.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
iooabslt.3 | ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) |
Ref | Expression |
---|---|
iooabslt | ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooabslt.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 11239 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | iooabslt.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) | |
4 | elioore 13351 | . . . . 5 ⊢ (𝐶 ∈ ((𝐴 − 𝐵)(,)(𝐴 + 𝐵)) → 𝐶 ∈ ℝ) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
6 | 5 | recnd 11239 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
7 | eqid 2724 | . . . 4 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
8 | 7 | cnmetdval 24609 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴(abs ∘ − )𝐶) = (abs‘(𝐴 − 𝐶))) |
9 | 2, 6, 8 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐴(abs ∘ − )𝐶) = (abs‘(𝐴 − 𝐶))) |
10 | iooabslt.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
11 | eqid 2724 | . . . . . . . . . 10 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
12 | 11 | bl2ioo 24630 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵) = ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) |
13 | 1, 10, 12 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵) = ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) |
14 | 3, 13 | eleqtrrd 2828 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵)) |
15 | cnxmet 24611 | . . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
16 | 15 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ)) |
17 | 2, 1 | elind 4186 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∩ ℝ)) |
18 | 10 | rexrd 11261 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
19 | 11 | blres 24259 | . . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ (ℂ ∩ ℝ) ∧ 𝐵 ∈ ℝ*) → (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵) = ((𝐴(ball‘(abs ∘ − ))𝐵) ∩ ℝ)) |
20 | 16, 17, 18, 19 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵) = ((𝐴(ball‘(abs ∘ − ))𝐵) ∩ ℝ)) |
21 | 14, 20 | eleqtrd 2827 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((𝐴(ball‘(abs ∘ − ))𝐵) ∩ ℝ)) |
22 | elin 3956 | . . . . . 6 ⊢ (𝐶 ∈ ((𝐴(ball‘(abs ∘ − ))𝐵) ∩ ℝ) ↔ (𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵) ∧ 𝐶 ∈ ℝ)) | |
23 | 21, 22 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵) ∧ 𝐶 ∈ ℝ)) |
24 | 23 | simpld 494 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵)) |
25 | elbl 24216 | . . . . 5 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵) ↔ (𝐶 ∈ ℂ ∧ (𝐴(abs ∘ − )𝐶) < 𝐵))) | |
26 | 16, 2, 18, 25 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵) ↔ (𝐶 ∈ ℂ ∧ (𝐴(abs ∘ − )𝐶) < 𝐵))) |
27 | 24, 26 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ (𝐴(abs ∘ − )𝐶) < 𝐵)) |
28 | 27 | simprd 495 | . 2 ⊢ (𝜑 → (𝐴(abs ∘ − )𝐶) < 𝐵) |
29 | 9, 28 | eqbrtrrd 5162 | 1 ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∩ cin 3939 class class class wbr 5138 × cxp 5664 ↾ cres 5668 ∘ ccom 5670 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 ℝcr 11105 + caddc 11109 ℝ*cxr 11244 < clt 11245 − cmin 11441 (,)cioo 13321 abscabs 15178 ∞Metcxmet 21213 ballcbl 21215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-xadd 13090 df-ioo 13325 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 |
This theorem is referenced by: lptre2pt 44841 |
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