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| Mirrors > Home > MPE Home > Th. List > distspace | Structured version Visualization version GIF version | ||
| Description: A set π together with a (distance) function π· which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set π equipped with a distance π·, which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.) |
| Ref | Expression |
|---|---|
| distspace | β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β ((π·:(π Γ π)βΆβ* β§ (π΄π·π΄) = 0) β§ (0 β€ (π΄π·π΅) β§ (π΄π·π΅) = (π΅π·π΄)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | psmetf 24240 | . . . 4 β’ (π· β (PsMetβπ) β π·:(π Γ π)βΆβ*) | |
| 2 | 1 | 3ad2ant1 1130 | . . 3 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β π·:(π Γ π)βΆβ*) |
| 3 | psmet0 24242 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΄ β π) β (π΄π·π΄) = 0) | |
| 4 | 3 | 3adant3 1129 | . . 3 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΄) = 0) |
| 5 | 2, 4 | jca 510 | . 2 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π·:(π Γ π)βΆβ* β§ (π΄π·π΄) = 0)) |
| 6 | psmetge0 24246 | . 2 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β 0 β€ (π΄π·π΅)) | |
| 7 | psmetsym 24244 | . 2 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (π΅π·π΄)) | |
| 8 | 5, 6, 7 | jca32 514 | 1 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β ((π·:(π Γ π)βΆβ* β§ (π΄π·π΄) = 0) β§ (0 β€ (π΄π·π΅) β§ (π΄π·π΅) = (π΅π·π΄)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5152 Γ cxp 5680 βΆwf 6549 βcfv 6553 (class class class)co 7426 0cc0 11148 β*cxr 11287 β€ cle 11289 PsMetcpsmet 21277 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-2 12315 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-psmet 21285 |
| This theorem is referenced by: (None) |
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