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| Mirrors > Home > MPE Home > Th. List > mstri | 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 |
|---|---|
| mstri | β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) + (πΆπ·π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mscl.x | . . . 4 β’ π = (Baseβπ) | |
| 2 | mscl.d | . . . 4 β’ π· = (distβπ) | |
| 3 | 1, 2 | msmet2 23957 | . . 3 β’ (π β MetSp β (π· βΎ (π Γ π)) β (Metβπ)) |
| 4 | mettri 23849 | . . 3 β’ (((π· βΎ (π Γ π)) β (Metβπ) 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) + (πΆ(π· βΎ (π Γ π))π΅))) | |
| 5 | 3, 4 | sylan 580 | . 2 β’ ((π β MetSp 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) + (πΆ(π· βΎ (π Γ π))π΅))) |
| 6 | simpr1 1194 | . . 3 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 7 | simpr2 1195 | . . 3 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 8 | 6, 7 | ovresd 7570 | . 2 β’ ((π β MetSp 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 MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 10 | 6, 9 | ovresd 7570 | . . 3 β’ ((π β MetSp 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 7570 | . . 3 β’ ((π β MetSp 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 7423 | . 2 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄(π· βΎ (π Γ π))πΆ) + (πΆ(π· βΎ (π Γ π))π΅)) = ((π΄π·πΆ) + (πΆπ·π΅))) |
| 13 | 5, 8, 12 | 3brtr3d 5178 | 1 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄π·π΅) β€ ((π΄π·πΆ) + (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 5147 Γ cxp 5673 βΎ cres 5677 βcfv 6540 (class class class)co 7405 + caddc 11109 β€ cle 11245 Basecbs 17140 distcds 17202 Metcmet 20922 MetSpcms 23815 |
| 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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
| 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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-xms 23817 df-ms 23818 |
| This theorem is referenced by: ngptgp 24136 nlmvscnlem2 24193 |
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