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Mirrors > Home > MPE Home > Th. List > psmetf | Structured version Visualization version GIF version |
Description: The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
Ref | Expression |
---|---|
psmetf | β’ (π· β (PsMetβπ) β π·:(π Γ π)βΆβ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6884 | . . . 4 β’ (π· β (PsMetβπ) β π β V) | |
2 | ispsmet 23680 | . . . 4 β’ (π β V β (π· β (PsMetβπ) β (π·:(π Γ π)βΆβ* β§ βπ β π ((ππ·π) = 0 β§ βπ β π βπ β π (ππ·π) β€ ((ππ·π) +π (ππ·π)))))) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π· β (PsMetβπ) β (π· β (PsMetβπ) β (π·:(π Γ π)βΆβ* β§ βπ β π ((ππ·π) = 0 β§ βπ β π βπ β π (ππ·π) β€ ((ππ·π) +π (ππ·π)))))) |
4 | 3 | ibi 267 | . 2 β’ (π· β (PsMetβπ) β (π·:(π Γ π)βΆβ* β§ βπ β π ((ππ·π) = 0 β§ βπ β π βπ β π (ππ·π) β€ ((ππ·π) +π (ππ·π))))) |
5 | 4 | simpld 496 | 1 β’ (π· β (PsMetβπ) β π·:(π Γ π)βΆβ*) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3447 class class class wbr 5109 Γ cxp 5635 βΆwf 6496 βcfv 6500 (class class class)co 7361 0cc0 11059 β*cxr 11196 β€ cle 11198 +π cxad 13039 PsMetcpsmet 20803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 df-xr 11201 df-psmet 20811 |
This theorem is referenced by: psmetcl 23683 psmetxrge0 23689 psmetres2 23690 distspace 23692 metustss 23930 metustid 23933 metustsym 23934 metustexhalf 23935 metustfbas 23936 cfilucfil 23938 blval2 23941 metuel2 23944 restmetu 23949 metideq 32538 pstmxmet 32542 |
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