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| Mirrors > Home > MPE Home > Th. List > xmstri | Structured version Visualization version GIF version | ||
| Description: Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| mscl.x | β’ π = (Baseβπ) |
| mscl.d | β’ π· = (distβπ) |
| Ref | Expression |
|---|---|
| xmstri | β’ ((π β βMetSp 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 | mscl.x | . . . 4 β’ π = (Baseβπ) | |
| 2 | mscl.d | . . . 4 β’ π· = (distβπ) | |
| 3 | 1, 2 | xmsxmet2 23891 | . . 3 β’ (π β βMetSp β (π· βΎ (π Γ π)) β (βMetβπ)) |
| 4 | xmettri 23783 | . . 3 β’ (((π· βΎ (π Γ π)) β (βMetβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄(π· βΎ (π Γ π))π΅) β€ ((π΄(π· βΎ (π Γ π))πΆ) +π (πΆ(π· βΎ (π Γ π))π΅))) | |
| 5 | 3, 4 | sylan 580 | . 2 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄(π· βΎ (π Γ π))π΅) β€ ((π΄(π· βΎ (π Γ π))πΆ) +π (πΆ(π· βΎ (π Γ π))π΅))) |
| 6 | simpr1 1194 | . . 3 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 7 | simpr2 1195 | . . 3 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 8 | 6, 7 | ovresd 7556 | . 2 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) |
| 9 | simpr3 1196 | . . . 4 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 10 | 6, 9 | ovresd 7556 | . . 3 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄(π· βΎ (π Γ π))πΆ) = (π΄π·πΆ)) |
| 11 | 9, 7 | ovresd 7556 | . . 3 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (πΆ(π· βΎ (π Γ π))π΅) = (πΆπ·π΅)) |
| 12 | 10, 11 | oveq12d 7410 | . 2 β’ ((π β βMetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄(π· βΎ (π Γ π))πΆ) +π (πΆ(π· βΎ (π Γ π))π΅)) = ((π΄π·πΆ) +π (πΆπ·π΅))) |
| 13 | 5, 8, 12 | 3brtr3d 5171 | 1 β’ ((π β βMetSp 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 5140 Γ cxp 5666 βΎ cres 5670 βcfv 6531 (class class class)co 7392 β€ cle 11230 +π cxad 13071 Basecbs 17125 distcds 17187 βMetcxmet 20860 βMetSpcxms 23749 |
| 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 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-cnex 11147 ax-resscn 11148 ax-1cn 11149 ax-icn 11150 ax-addcl 11151 ax-addrcl 11152 ax-mulcl 11153 ax-mulrcl 11154 ax-mulcom 11155 ax-addass 11156 ax-mulass 11157 ax-distr 11158 ax-i2m1 11159 ax-1ne0 11160 ax-1rid 11161 ax-rnegex 11162 ax-rrecex 11163 ax-cnre 11164 ax-pre-lttri 11165 ax-pre-lttrn 11166 ax-pre-ltadd 11167 ax-pre-mulgt0 11168 ax-pre-sup 11169 |
| 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-rmo 3375 df-reu 3376 df-rab 3432 df-v 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7348 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7838 df-1st 7956 df-2nd 7957 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-rdg 8391 df-er 8685 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9418 df-inf 9419 df-pnf 11231 df-mnf 11232 df-xr 11233 df-ltxr 11234 df-le 11235 df-sub 11427 df-neg 11428 df-div 11853 df-nn 12194 df-2 12256 df-n0 12454 df-z 12540 df-uz 12804 df-q 12914 df-rp 12956 df-xneg 13073 df-xadd 13074 df-xmul 13075 df-topgen 17370 df-psmet 20867 df-xmet 20868 df-bl 20870 df-mopn 20871 df-top 22322 df-topon 22339 df-topsp 22361 df-bases 22375 df-xms 23752 |
| This theorem is referenced by: (None) |
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