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| Mirrors > Home > MPE Home > Th. List > mstri2 | Structured version Visualization version GIF version | ||
| Description: Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
| Ref | Expression |
|---|---|
| mscl.x | β’ π = (Baseβπ) |
| mscl.d | β’ π· = (distβπ) |
| Ref | Expression |
|---|---|
| mstri2 | β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) + (πΆπ·π΅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mscl.x | . . . 4 β’ π = (Baseβπ) | |
| 2 | mscl.d | . . . 4 β’ π· = (distβπ) | |
| 3 | 1, 2 | msmet2 23966 | . . 3 β’ (π β MetSp β (π· βΎ (π Γ π)) β (Metβπ)) |
| 4 | mettri2 23847 | . . 3 β’ (((π· βΎ (π Γ π)) β (Metβπ) β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄(π· βΎ (π Γ π))π΅) β€ ((πΆ(π· βΎ (π Γ π))π΄) + (πΆ(π· βΎ (π Γ π))π΅))) | |
| 5 | 3, 4 | sylan 581 | . 2 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄(π· βΎ (π Γ π))π΅) β€ ((πΆ(π· βΎ (π Γ π))π΄) + (πΆ(π· βΎ (π Γ π))π΅))) |
| 6 | simpr2 1196 | . . 3 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β π΄ β π) | |
| 7 | simpr3 1197 | . . 3 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β π΅ β π) | |
| 8 | 6, 7 | ovresd 7574 | . 2 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) |
| 9 | simpr1 1195 | . . . 4 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β πΆ β π) | |
| 10 | 9, 6 | ovresd 7574 | . . 3 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆ(π· βΎ (π Γ π))π΄) = (πΆπ·π΄)) |
| 11 | 9, 7 | ovresd 7574 | . . 3 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (πΆ(π· βΎ (π Γ π))π΅) = (πΆπ·π΅)) |
| 12 | 10, 11 | oveq12d 7427 | . 2 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β ((πΆ(π· βΎ (π Γ π))π΄) + (πΆ(π· βΎ (π Γ π))π΅)) = ((πΆπ·π΄) + (πΆπ·π΅))) |
| 13 | 5, 8, 12 | 3brtr3d 5180 | 1 β’ ((π β MetSp β§ (πΆ β π β§ π΄ β π β§ π΅ β π)) β (π΄π·π΅) β€ ((πΆπ·π΄) + (πΆπ·π΅))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5149 Γ cxp 5675 βΎ cres 5679 βcfv 6544 (class class class)co 7409 + caddc 11113 β€ cle 11249 Basecbs 17144 distcds 17206 Metcmet 20930 MetSpcms 23824 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-xms 23826 df-ms 23827 |
| This theorem is referenced by: (None) |
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