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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 24167 | . . . 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 24169 | . . . 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 511 | . 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 24173 | . 2 β’ ((π· β (PsMetβπ) β§ π΄ β π β§ π΅ β π) β 0 β€ (π΄π·π΅)) | |
| 7 | psmetsym 24171 | . 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 515 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 Γ cxp 5667 βΆwf 6533 βcfv 6537 (class class class)co 7405 0cc0 11112 β*cxr 11251 β€ cle 11253 PsMetcpsmet 21224 |
| 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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-2 12279 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-psmet 21232 |
| This theorem is referenced by: (None) |
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