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| Mirrors > Home > MPE Home > Th. List > psmettri | Structured version Visualization version GIF version | ||
| Description: Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| Ref | Expression |
|---|---|
| psmettri | β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) +π (πΆπ·π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 481 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π· β (PsMetβπ)) | |
| 2 | simpr3 1193 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 3 | simpr1 1191 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 4 | simpr2 1192 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 5 | psmettri2 24231 | . . 3 β’ ((π· β (PsMetβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +π (πΆπ·π΅))) | |
| 6 | 1, 2, 3, 4, 5 | syl13anc 1369 | . 2 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) +π (πΆπ·π΅))) |
| 7 | psmetsym 24232 | . . . 4 β’ ((π· β (PsMetβπ) β§ πΆ β π β§ π΄ β π) β (πΆπ·π΄) = (π΄π·πΆ)) | |
| 8 | 1, 2, 3, 7 | syl3anc 1368 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πΆπ·π΄) = (π΄π·πΆ)) |
| 9 | 8 | oveq1d 7429 | . 2 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((πΆπ·π΄) +π (πΆπ·π΅)) = ((π΄π·πΆ) +π (πΆπ·π΅))) |
| 10 | 6, 9 | breqtrd 5167 | 1 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) +π (πΆπ·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 5141 βcfv 6541 (class class class)co 7414 β€ cle 11277 +π cxad 13120 PsMetcpsmet 21265 |
| 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 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 |
| 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-po 5582 df-so 5583 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-xadd 13123 df-psmet 21273 |
| This theorem is referenced by: metustexhalf 24481 |
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