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Mirrors > Home > MPE Home > Th. List > metf | Structured version Visualization version GIF version |
Description: Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) |
Ref | Expression |
---|---|
metf | ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metflem 22932 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))) | |
2 | 1 | simpld 497 | 1 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5059 × cxp 5548 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 + caddc 10534 ≤ cle 10670 Metcmet 20525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-met 20533 |
This theorem is referenced by: metcl 22936 metn0 22964 metres2 22967 metres 22969 msf 23062 isngp3 23201 tngngp2 23255 tngngpim 23262 xrsdsre 23412 metdcn2 23441 cncms 23952 cnrrext 31246 isbnd3 35056 isbnd3b 35057 ssbnd 35060 bnd2lem 35063 prdsbnd 35065 |
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