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Mirrors > Home > MPE Home > Th. List > metdsf | Structured version Visualization version GIF version |
Description: The distance from a point to a set is a nonnegative extended real number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.) |
Ref | Expression |
---|---|
metdscn.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) |
Ref | Expression |
---|---|
metdsf | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplll 815 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | simplr 809 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑋) | |
3 | simplr 809 | . . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑆 ⊆ 𝑋) | |
4 | 3 | sselda 3677 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑋) |
5 | xmetcl 22226 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*) | |
6 | 1, 2, 4, 5 | syl3anc 1407 | . . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → (𝑥𝐷𝑦) ∈ ℝ*) |
7 | eqid 2692 | . . . . . 6 ⊢ (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) | |
8 | 6, 7 | fmptd 6468 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)):𝑆⟶ℝ*) |
9 | frn 6134 | . . . . 5 ⊢ ((𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)):𝑆⟶ℝ* → ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) ⊆ ℝ*) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) ⊆ ℝ*) |
11 | infxrcl 12245 | . . . 4 ⊢ (ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) ⊆ ℝ* → inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ ℝ*) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ ℝ*) |
13 | xmetge0 22239 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐷𝑦)) | |
14 | 1, 2, 4, 13 | syl3anc 1407 | . . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 0 ≤ (𝑥𝐷𝑦)) |
15 | 14 | ralrimiva 3036 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑆 0 ≤ (𝑥𝐷𝑦)) |
16 | ovex 6761 | . . . . . . 7 ⊢ (𝑥𝐷𝑦) ∈ V | |
17 | 16 | rgenw 2994 | . . . . . 6 ⊢ ∀𝑦 ∈ 𝑆 (𝑥𝐷𝑦) ∈ V |
18 | breq2 4732 | . . . . . . 7 ⊢ (𝑧 = (𝑥𝐷𝑦) → (0 ≤ 𝑧 ↔ 0 ≤ (𝑥𝐷𝑦))) | |
19 | 7, 18 | ralrnmpt 6451 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑆 (𝑥𝐷𝑦) ∈ V → (∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 0 ≤ (𝑥𝐷𝑦))) |
20 | 17, 19 | ax-mp 5 | . . . . 5 ⊢ (∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 0 ≤ (𝑥𝐷𝑦)) |
21 | 15, 20 | sylibr 224 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧) |
22 | 0xr 10167 | . . . . 5 ⊢ 0 ∈ ℝ* | |
23 | infxrgelb 12247 | . . . . 5 ⊢ ((ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) ⊆ ℝ* ∧ 0 ∈ ℝ*) → (0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧)) | |
24 | 10, 22, 23 | sylancl 697 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧)) |
25 | 21, 24 | mpbird 247 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → 0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) |
26 | elxrge0 12363 | . . 3 ⊢ (inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ (0[,]+∞) ↔ (inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))) | |
27 | 12, 25, 26 | sylanbrc 701 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ (0[,]+∞)) |
28 | metdscn.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
29 | 27, 28 | fmptd 6468 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1564 ∈ wcel 2071 ∀wral 2982 Vcvv 3272 ⊆ wss 3648 class class class wbr 4728 ↦ cmpt 4805 ran crn 5187 ⟶wf 5965 ‘cfv 5969 (class class class)co 6733 infcinf 8431 0cc0 10017 +∞cpnf 10152 ℝ*cxr 10154 < clt 10155 ≤ cle 10156 [,]cicc 12260 ∞Metcxmt 19822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 ax-pre-sup 10095 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-op 4260 df-uni 4513 df-iun 4598 df-br 4729 df-opab 4789 df-mpt 4806 df-id 5096 df-po 5107 df-so 5108 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-1st 7253 df-2nd 7254 df-er 7830 df-map 7944 df-en 8041 df-dom 8042 df-sdom 8043 df-sup 8432 df-inf 8433 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-div 10766 df-2 11160 df-rp 11915 df-xneg 12028 df-xadd 12029 df-xmul 12030 df-icc 12264 df-xmet 19830 |
This theorem is referenced by: metds0 22743 metdstri 22744 metdsre 22746 metdseq0 22747 metdscnlem 22748 metdscn 22749 metnrmlem1a 22751 metnrmlem1 22752 lebnumlem1 22850 |
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