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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 794 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | simplr 788 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑋) | |
3 | simplr 788 | . . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑆 ⊆ 𝑋) | |
4 | 3 | sselda 3568 | . . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑋) |
5 | xmetcl 21946 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*) | |
6 | 1, 2, 4, 5 | syl3anc 1318 | . . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → (𝑥𝐷𝑦) ∈ ℝ*) |
7 | eqid 2610 | . . . . . 6 ⊢ (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) = (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)) | |
8 | 6, 7 | fmptd 6292 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)):𝑆⟶ℝ*) |
9 | frn 5966 | . . . . 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 12035 | . . . 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 21959 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐷𝑦)) | |
14 | 1, 2, 4, 13 | syl3anc 1318 | . . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑆) → 0 ≤ (𝑥𝐷𝑦)) |
15 | 14 | ralrimiva 2949 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑆 0 ≤ (𝑥𝐷𝑦)) |
16 | ovex 6577 | . . . . . . 7 ⊢ (𝑥𝐷𝑦) ∈ V | |
17 | 16 | rgenw 2908 | . . . . . 6 ⊢ ∀𝑦 ∈ 𝑆 (𝑥𝐷𝑦) ∈ V |
18 | breq2 4587 | . . . . . . 7 ⊢ (𝑧 = (𝑥𝐷𝑦) → (0 ≤ 𝑧 ↔ 0 ≤ (𝑥𝐷𝑦))) | |
19 | 7, 18 | ralrnmpt 6276 | . . . . . 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 223 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦))0 ≤ 𝑧) |
22 | 0xr 9965 | . . . . 5 ⊢ 0 ∈ ℝ* | |
23 | infxrgelb 12037 | . . . . 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 693 | . . . 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 246 | . . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑋) → 0 ≤ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) |
26 | elxrge0 12152 | . . 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 695 | . 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 6292 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 infcinf 8230 0cc0 9815 +∞cpnf 9950 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,]cicc 12049 ∞Metcxmt 19552 |
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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 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-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-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 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-2 10956 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-xmet 19560 |
This theorem is referenced by: metds0 22461 metdstri 22462 metdsre 22464 metdseq0 22465 metdscnlem 22466 metdscn 22467 metnrmlem1a 22469 metnrmlem1 22470 lebnumlem1 22568 |
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