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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 9947 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | iooabslt.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) | |
4 | elioore 12076 | . . . . 5 ⊢ (𝐶 ∈ ((𝐴 − 𝐵)(,)(𝐴 + 𝐵)) → 𝐶 ∈ ℝ) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
6 | 5 | recnd 9947 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
7 | eqid 2610 | . . . 4 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
8 | 7 | cnmetdval 22384 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴(abs ∘ − )𝐶) = (abs‘(𝐴 − 𝐶))) |
9 | 2, 6, 8 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐴(abs ∘ − )𝐶) = (abs‘(𝐴 − 𝐶))) |
10 | iooabslt.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
11 | eqid 2610 | . . . . . . . . . 10 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
12 | 11 | bl2ioo 22403 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵) = ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) |
13 | 1, 10, 12 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵) = ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) |
14 | 3, 13 | eleqtrrd 2691 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵)) |
15 | cnxmet 22386 | . . . . . . . . 9 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
16 | 15 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ)) |
17 | 2, 1 | elind 3760 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∩ ℝ)) |
18 | 10 | rexrd 9968 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
19 | 11 | blres 22046 | . . . . . . . 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 1318 | . . . . . . 7 ⊢ (𝜑 → (𝐴(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝐵) = ((𝐴(ball‘(abs ∘ − ))𝐵) ∩ ℝ)) |
21 | 14, 20 | eleqtrd 2690 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ((𝐴(ball‘(abs ∘ − ))𝐵) ∩ ℝ)) |
22 | elin 3758 | . . . . . 6 ⊢ (𝐶 ∈ ((𝐴(ball‘(abs ∘ − ))𝐵) ∩ ℝ) ↔ (𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵) ∧ 𝐶 ∈ ℝ)) | |
23 | 21, 22 | sylib 207 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵) ∧ 𝐶 ∈ ℝ)) |
24 | 23 | simpld 474 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵)) |
25 | elbl 22003 | . . . . 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 1318 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐴(ball‘(abs ∘ − ))𝐵) ↔ (𝐶 ∈ ℂ ∧ (𝐴(abs ∘ − )𝐶) < 𝐵))) |
27 | 24, 26 | mpbid 221 | . . 3 ⊢ (𝜑 → (𝐶 ∈ ℂ ∧ (𝐴(abs ∘ − )𝐶) < 𝐵)) |
28 | 27 | simprd 478 | . 2 ⊢ (𝜑 → (𝐴(abs ∘ − )𝐶) < 𝐵) |
29 | 9, 28 | eqbrtrrd 4607 | 1 ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 class class class wbr 4583 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 + caddc 9818 ℝ*cxr 9952 < clt 9953 − cmin 10145 (,)cioo 12046 abscabs 13822 ∞Metcxmt 19552 ballcbl 19554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-ioo 12050 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 |
This theorem is referenced by: lptre2pt 38707 |
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