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Mirrors > Home > ILE Home > Th. List > bdmet | GIF version |
Description: The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.) |
Ref | Expression |
---|---|
stdbdmet.1 | ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < )) |
Ref | Expression |
---|---|
bdmet | ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpxr 9442 | . . . 4 ⊢ (𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ*) | |
2 | rpgt0 9446 | . . . 4 ⊢ (𝑅 ∈ ℝ+ → 0 < 𝑅) | |
3 | 1, 2 | jca 304 | . . 3 ⊢ (𝑅 ∈ ℝ+ → (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) |
4 | stdbdmet.1 | . . . . 5 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < )) | |
5 | 4 | bdxmet 12659 | . . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋)) |
6 | 5 | 3expb 1182 | . . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝐷 ∈ (∞Met‘𝑋)) |
7 | 3, 6 | sylan2 284 | . 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (∞Met‘𝑋)) |
8 | xmetcl 12510 | . . . . . . . 8 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐶𝑦) ∈ ℝ*) | |
9 | 8 | 3expb 1182 | . . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ ℝ*) |
10 | 9 | adantlr 468 | . . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ∈ ℝ*) |
11 | 1 | ad2antlr 480 | . . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑅 ∈ ℝ*) |
12 | xrmincl 11028 | . . . . . 6 ⊢ (((𝑥𝐶𝑦) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ∈ ℝ*) | |
13 | 10, 11, 12 | syl2anc 408 | . . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ∈ ℝ*) |
14 | rpre 9441 | . . . . . 6 ⊢ (𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ) | |
15 | 14 | ad2antlr 480 | . . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑅 ∈ ℝ) |
16 | xmetge0 12523 | . . . . . . . 8 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐶𝑦)) | |
17 | 16 | 3expb 1182 | . . . . . . 7 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ (𝑥𝐶𝑦)) |
18 | 17 | adantlr 468 | . . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ (𝑥𝐶𝑦)) |
19 | rpge0 9447 | . . . . . . 7 ⊢ (𝑅 ∈ ℝ+ → 0 ≤ 𝑅) | |
20 | 19 | ad2antlr 480 | . . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ 𝑅) |
21 | 0xr 7805 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
22 | xrlemininf 11033 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ (𝑥𝐶𝑦) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → (0 ≤ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ↔ (0 ≤ (𝑥𝐶𝑦) ∧ 0 ≤ 𝑅))) | |
23 | 21, 10, 11, 22 | mp3an2i 1320 | . . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (0 ≤ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ↔ (0 ≤ (𝑥𝐶𝑦) ∧ 0 ≤ 𝑅))) |
24 | 18, 20, 23 | mpbir2and 928 | . . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < )) |
25 | xrmin2inf 11030 | . . . . . 6 ⊢ (((𝑥𝐶𝑦) ∈ ℝ* ∧ 𝑅 ∈ ℝ*) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ≤ 𝑅) | |
26 | 10, 11, 25 | syl2anc 408 | . . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ≤ 𝑅) |
27 | xrrege0 9601 | . . . . 5 ⊢ (((inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ∈ ℝ* ∧ 𝑅 ∈ ℝ) ∧ (0 ≤ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ∧ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ≤ 𝑅)) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ∈ ℝ) | |
28 | 13, 15, 24, 26, 27 | syl22anc 1217 | . . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ∈ ℝ) |
29 | 28 | ralrimivva 2512 | . . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ∈ ℝ) |
30 | 4 | fmpo 6092 | . . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ) ∈ ℝ ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ) |
31 | 29, 30 | sylib 121 | . 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
32 | ismet2 12512 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷:(𝑋 × 𝑋)⟶ℝ)) | |
33 | 7, 31, 32 | sylanbrc 413 | 1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∀wral 2414 {cpr 3523 class class class wbr 3924 × cxp 4532 ⟶wf 5114 ‘cfv 5118 (class class class)co 5767 ∈ cmpo 5769 infcinf 6863 ℝcr 7612 0cc0 7613 ℝ*cxr 7792 < clt 7793 ≤ cle 7794 ℝ+crp 9434 ∞Metcxmet 12138 Metcmet 12139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-map 6537 df-sup 6864 df-inf 6865 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-rp 9435 df-xneg 9552 df-xadd 9553 df-icc 9671 df-seqfrec 10212 df-exp 10286 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-xmet 12146 df-met 12147 |
This theorem is referenced by: mopnex 12663 |
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