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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 483 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π· β (PsMetβπ)) | |
| 2 | simpr3 1196 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 3 | simpr1 1194 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 4 | simpr2 1195 | . . 3 β’ ((π· β (PsMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 5 | psmettri2 23739 | . . 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 1372 | . 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 23740 | . . . 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 1371 | . . 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 7405 | . 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 5164 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5138 βcfv 6529 (class class class)co 7390 β€ cle 11228 +π cxad 13069 PsMetcpsmet 20857 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8683 df-map 8802 df-en 8920 df-dom 8921 df-sdom 8922 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-xadd 13072 df-psmet 20865 |
| This theorem is referenced by: metustexhalf 23989 |
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