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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 11241 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 + 𝐴) ∈ ℝ) | |
2 | 1 | ancoms 457 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 + 𝐴) ∈ ℝ) |
3 | 2 | rehalfcld 12511 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + 𝐴) / 2) ∈ ℝ) |
4 | resubcl 11574 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) | |
5 | 4 | ancoms 457 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) |
6 | 5 | rehalfcld 12511 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 − 𝐴) / 2) ∈ ℝ) |
7 | remet.1 | . . . 4 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
8 | 7 | bl2ioo 24799 | . . 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 582 | . 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 11248 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
11 | recn 11248 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
12 | addcom 11450 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) | |
13 | 10, 11, 12 | syl2anr 595 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) |
14 | 13 | oveq1d 7439 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐵 + 𝐴) / 2) = ((𝐴 + 𝐵) / 2)) |
15 | 14 | oveq1d 7439 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐵 + 𝐴) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2)) = (((𝐴 + 𝐵) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2))) |
16 | halfaddsub 12497 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2)) = 𝐵 ∧ (((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2)) = 𝐴)) | |
17 | 10, 11, 16 | syl2anr 595 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2)) = 𝐵 ∧ (((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2)) = 𝐴)) |
18 | 17 | simprd 494 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2)) = 𝐴) |
19 | 17 | simpld 493 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2)) = 𝐵) |
20 | 18, 19 | oveq12d 7442 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((((𝐵 + 𝐴) / 2) − ((𝐵 − 𝐴) / 2))(,)(((𝐵 + 𝐴) / 2) + ((𝐵 − 𝐴) / 2))) = (𝐴(,)𝐵)) |
21 | 9, 15, 20 | 3eqtr3rd 2775 | 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 394 = wceq 1534 ∈ wcel 2099 × cxp 5680 ↾ cres 5684 ∘ ccom 5686 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 + caddc 11161 − cmin 11494 / cdiv 11921 2c2 12319 (,)cioo 13378 abscabs 15239 ballcbl 21330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-xadd 13147 df-ioo 13382 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 |
This theorem is referenced by: ioo2blex 24801 |
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