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Mirrors > Home > MPE Home > Th. List > xmetpsmet | Structured version Visualization version GIF version |
Description: An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
Ref | Expression |
---|---|
xmetpsmet | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmetf 22459 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | xmet0 22472 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷𝑥) = 0) | |
3 | 3anrot 1123 | . . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) | |
4 | xmettri2 22470 | . . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) | |
5 | 3, 4 | sylan2br 589 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
6 | 5 | 3anassrs 1470 | . . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
7 | 6 | ralrimiva 3145 | . . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
8 | 7 | ralrimiva 3145 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
9 | 2, 8 | jca 508 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) |
10 | 9 | ralrimiva 3145 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) |
11 | elfvex 6443 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ V) | |
12 | ispsmet 22434 | . . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) |
14 | 1, 10, 13 | mpbir2and 705 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∀wral 3087 Vcvv 3383 class class class wbr 4841 × cxp 5308 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 0cc0 10222 ℝ*cxr 10360 ≤ cle 10362 +𝑒 cxad 12187 PsMetcpsmet 20049 ∞Metcxmet 20050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-map 8095 df-xr 10365 df-psmet 20057 df-xmet 20058 |
This theorem is referenced by: blfval 22514 xmetutop 22698 xmsusp 22699 cfilucfil3 23443 cmetcusp 23477 cnflduss 23479 reust 23504 qqhucn 30544 sitmcl 30921 heicant 33925 metpsmet 40015 ioorrnopnlem 41255 |
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