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Mirrors > Home > MPE Home > Th. List > ioo2bl | Structured version Visualization version GIF version |
Description: An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
remet.1 | ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) |
Ref | Expression |
---|---|
ioo2bl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(,)𝐵) = (((𝐴 + 𝐵) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | readdcl 11269 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 + 𝐴) ∈ ℝ) | |
2 | 1 | ancoms 458 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 + 𝐴) ∈ ℝ) |
3 | 2 | rehalfcld 12542 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + 𝐴) / 2) ∈ ℝ) |
4 | resubcl 11602 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) | |
5 | 4 | ancoms 458 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) |
6 | 5 | rehalfcld 12542 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 − 𝐴) / 2) ∈ ℝ) |
7 | remet.1 | . . . 4 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
8 | 7 | bl2ioo 24835 | . . 3 ⊢ ((((𝐵 + 𝐴) / 2) ∈ ℝ ∧ ((𝐵 − 𝐴) / 2) ∈ ℝ) → (((𝐵 + 𝐴) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2)) = ((((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2))(,)(((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2)))) |
9 | 3, 6, 8 | syl2anc 583 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐵 + 𝐴) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2)) = ((((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2))(,)(((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2)))) |
10 | recn 11276 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | recn 11276 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
12 | addcom 11478 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) | |
13 | 10, 11, 12 | syl2anr 596 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) |
14 | 13 | oveq1d 7465 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + 𝐴) / 2) = ((𝐴 + 𝐵) / 2)) |
15 | 14 | oveq1d 7465 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐵 + 𝐴) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2)) = (((𝐴 + 𝐵) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2))) |
16 | halfaddsub 12528 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2)) = 𝐵 ∧ (((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2)) = 𝐴)) | |
17 | 10, 11, 16 | syl2anr 596 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2)) = 𝐵 ∧ (((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2)) = 𝐴)) |
18 | 17 | simprd 495 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2)) = 𝐴) |
19 | 17 | simpld 494 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2)) = 𝐵) |
20 | 18, 19 | oveq12d 7468 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2))(,)(((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2))) = (𝐴(,)𝐵)) |
21 | 9, 15, 20 | 3eqtr3rd 2789 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(,)𝐵) = (((𝐴 + 𝐵) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 × cxp 5698 ↾ cres 5702 ∘ ccom 5704 ‘cfv 6575 (class class class)co 7450 ℂcc 11184 ℝcr 11185 + caddc 11189 − cmin 11522 / cdiv 11949 2c2 12350 (,)cioo 13409 abscabs 15285 ballcbl 21376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-sup 9513 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-n0 12556 df-z 12642 df-uz 12906 df-rp 13060 df-xadd 13178 df-ioo 13413 df-seq 14055 df-exp 14115 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-psmet 21381 df-xmet 21382 df-met 21383 df-bl 21384 |
This theorem is referenced by: ioo2blex 24837 |
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