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Mirrors > Home > ILE Home > Th. List > bl2in | GIF version |
Description: Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
bl2in | ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑅)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1002 | . . 3 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → 𝐷 ∈ (Met‘𝑋)) | |
2 | metxmet 14332 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → 𝐷 ∈ (∞Met‘𝑋)) |
4 | simpl2 1003 | . 2 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → 𝑃 ∈ 𝑋) | |
5 | simpl3 1004 | . 2 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → 𝑄 ∈ 𝑋) | |
6 | rexr 8034 | . . 3 ⊢ (𝑅 ∈ ℝ → 𝑅 ∈ ℝ*) | |
7 | 6 | ad2antrl 490 | . 2 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → 𝑅 ∈ ℝ*) |
8 | simprl 529 | . . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → 𝑅 ∈ ℝ) | |
9 | rexadd 9884 | . . . . 5 ⊢ ((𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑅 +𝑒 𝑅) = (𝑅 + 𝑅)) | |
10 | 8, 8, 9 | syl2anc 411 | . . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → (𝑅 +𝑒 𝑅) = (𝑅 + 𝑅)) |
11 | 8 | recnd 8017 | . . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → 𝑅 ∈ ℂ) |
12 | 11 | 2timesd 9192 | . . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → (2 · 𝑅) = (𝑅 + 𝑅)) |
13 | 10, 12 | eqtr4d 2225 | . . 3 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → (𝑅 +𝑒 𝑅) = (2 · 𝑅)) |
14 | id 19 | . . . . . 6 ⊢ (𝑅 ∈ ℝ → 𝑅 ∈ ℝ) | |
15 | metcl 14330 | . . . . . 6 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ) | |
16 | 2re 9020 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
17 | 2pos 9041 | . . . . . . . 8 ⊢ 0 < 2 | |
18 | 16, 17 | pm3.2i 272 | . . . . . . 7 ⊢ (2 ∈ ℝ ∧ 0 < 2) |
19 | lemuldiv2 8870 | . . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ (𝑃𝐷𝑄) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · 𝑅) ≤ (𝑃𝐷𝑄) ↔ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) | |
20 | 18, 19 | mp3an3 1337 | . . . . . 6 ⊢ ((𝑅 ∈ ℝ ∧ (𝑃𝐷𝑄) ∈ ℝ) → ((2 · 𝑅) ≤ (𝑃𝐷𝑄) ↔ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) |
21 | 14, 15, 20 | syl2anr 290 | . . . . 5 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ 𝑅 ∈ ℝ) → ((2 · 𝑅) ≤ (𝑃𝐷𝑄) ↔ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) |
22 | 21 | biimprd 158 | . . . 4 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ 𝑅 ∈ ℝ) → (𝑅 ≤ ((𝑃𝐷𝑄) / 2) → (2 · 𝑅) ≤ (𝑃𝐷𝑄))) |
23 | 22 | impr 379 | . . 3 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → (2 · 𝑅) ≤ (𝑃𝐷𝑄)) |
24 | 13, 23 | eqbrtrd 4040 | . 2 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → (𝑅 +𝑒 𝑅) ≤ (𝑃𝐷𝑄)) |
25 | bldisj 14378 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑅 ∈ ℝ* ∧ (𝑅 +𝑒 𝑅) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑅)) = ∅) | |
26 | 3, 4, 5, 7, 7, 24, 25 | syl33anc 1264 | 1 ⊢ (((𝐷 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑅 ≤ ((𝑃𝐷𝑄) / 2))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑅)) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∩ cin 3143 ∅c0 3437 class class class wbr 4018 ‘cfv 5235 (class class class)co 5897 ℝcr 7841 0cc0 7842 + caddc 7845 · cmul 7847 ℝ*cxr 8022 < clt 8023 ≤ cle 8024 / cdiv 8660 2c2 9001 +𝑒 cxad 9802 ∞Metcxmet 13866 Metcmet 13867 ballcbl 13868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-map 6677 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-2 9009 df-xadd 9805 df-psmet 13873 df-xmet 13874 df-met 13875 df-bl 13876 |
This theorem is referenced by: (None) |
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