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Mirrors > Home > MPE Home > Th. List > xmet0 | Structured version Visualization version GIF version |
Description: The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of [Gleason] p. 223. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmet0 | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | xmeteq0 22948 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐷𝐴) = 0 ↔ 𝐴 = 𝐴)) | |
3 | 2 | 3anidm23 1417 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐷𝐴) = 0 ↔ 𝐴 = 𝐴)) |
4 | 1, 3 | mpbiri 260 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ∞Metcxmet 20530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-xr 10679 df-xmet 20538 |
This theorem is referenced by: met0 22953 xmetge0 22954 xmetsym 22957 xmetpsmet 22958 xblcntr 23021 ssbl 23033 xmeter 23043 ubthlem2 28648 sitmcl 31609 |
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