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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 773 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | simplr 767 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑋) | |
3 | simplr 767 | . . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑆 ⊆ 𝑋) | |
4 | 3 | sselda 3967 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑋) |
5 | xmetcl 22941 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*) | |
6 | 1, 2, 4, 5 | syl3anc 1367 | . . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → (𝑥𝐷𝑦) ∈ ℝ*) |
7 | eqid 2821 | . . . . . 6 ⊢ (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) | |
8 | 6, 7 | fmptd 6878 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)):𝑆⟶ℝ*) |
9 | 8 | frnd 6521 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) ⊆ ℝ*) |
10 | infxrcl 12727 | . . . 4 ⊢ (ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) ⊆ ℝ* → inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ ℝ*) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ ℝ*) |
12 | xmetge0 22954 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐷𝑦)) | |
13 | 1, 2, 4, 12 | syl3anc 1367 | . . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 0 ≤ (𝑥𝐷𝑦)) |
14 | 13 | ralrimiva 3182 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑆 0 ≤ (𝑥𝐷𝑦)) |
15 | ovex 7189 | . . . . . . 7 ⊢ (𝑥𝐷𝑦) ∈ V | |
16 | 15 | rgenw 3150 | . . . . . 6 ⊢ ∀𝑦 ∈ 𝑆 (𝑥𝐷𝑦) ∈ V |
17 | breq2 5070 | . . . . . . 7 ⊢ (𝑧 = (𝑥𝐷𝑦) → (0 ≤ 𝑧 ↔ 0 ≤ (𝑥𝐷𝑦))) | |
18 | 7, 17 | ralrnmptw 6860 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝑆 (𝑥𝐷𝑦) ∈ V → (∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 0 ≤ (𝑥𝐷𝑦))) |
19 | 16, 18 | ax-mp 5 | . . . . 5 ⊢ (∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 0 ≤ (𝑥𝐷𝑦)) |
20 | 14, 19 | sylibr 236 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧) |
21 | 0xr 10688 | . . . . 5 ⊢ 0 ∈ ℝ* | |
22 | infxrgelb 12729 | . . . . 5 ⊢ ((ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) ⊆ ℝ* ∧ 0 ∈ ℝ*) → (0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧)) | |
23 | 9, 21, 22 | sylancl 588 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ↔ ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧)) |
24 | 20, 23 | mpbird 259 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → 0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) |
25 | elxrge0 12846 | . . 3 ⊢ (inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ (0[,]+∞) ↔ (inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ ℝ* ∧ 0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))) | |
26 | 11, 24, 25 | sylanbrc 585 | . 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ) ∈ (0[,]+∞)) |
27 | metdscn.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
28 | 26, 27 | fmptd 6878 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 infcinf 8905 0cc0 10537 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 [,]cicc 12742 ∞Metcxmet 20530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-xmet 20538 |
This theorem is referenced by: metds0 23458 metdstri 23459 metdsre 23461 metdseq0 23462 metdscnlem 23463 metdscn 23464 metnrmlem1a 23466 metnrmlem1 23467 lebnumlem1 23565 |
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