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Mirrors > Home > MPE Home > Th. List > xmetf | Structured version Visualization version GIF version |
Description: Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmetf | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6469 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) | |
2 | isxmet 22506 | . . . 4 ⊢ (𝑋 ∈ dom ∞Met → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) |
4 | 3 | ibi 259 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))) |
5 | 4 | simpld 490 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 class class class wbr 4875 × cxp 5344 dom cdm 5346 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 0cc0 10259 ℝ*cxr 10397 ≤ cle 10399 +𝑒 cxad 12237 ∞Metcxmet 20098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-xr 10402 df-xmet 20106 |
This theorem is referenced by: xmetcl 22513 xmetdmdm 22517 xmetpsmet 22530 xmettpos 22531 xmetres2 22543 xmetres 22546 imasdsf1olem 22555 xmeterval 22614 xmeter 22615 xmetresbl 22619 tmsval 22663 tmslem 22664 tmsxms 22668 imasf1oxms 22671 comet 22695 stdbdxmet 22697 prdsxms 22712 xrsdsre 22990 xmetdcn2 23017 iscfil2 23441 caufval 23450 isbndx 34122 ssbnd 34128 ismtyval 34140 |
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