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| Mirrors > Home > MPE Home > Th. List > msrtri | Structured version Visualization version GIF version | ||
| Description: Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| Ref | Expression |
|---|---|
| mscl.x | β’ π = (Baseβπ) |
| mscl.d | β’ π· = (distβπ) |
| Ref | Expression |
|---|---|
| msrtri | β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (absβ((π΄π·πΆ) β (π΅π·πΆ))) β€ (π΄π·π΅)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mscl.x | . . . 4 β’ π = (Baseβπ) | |
| 2 | mscl.d | . . . 4 β’ π· = (distβπ) | |
| 3 | 1, 2 | msmet2 24379 | . . 3 β’ (π β MetSp β (π· βΎ (π Γ π)) β (Metβπ)) |
| 4 | metrtri 24276 | . . 3 β’ (((π· βΎ (π Γ π)) β (Metβπ) β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (absβ((π΄(π· βΎ (π Γ π))πΆ) β (π΅(π· βΎ (π Γ π))πΆ))) β€ (π΄(π· βΎ (π Γ π))π΅)) | |
| 5 | 3, 4 | sylan 579 | . 2 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (absβ((π΄(π· βΎ (π Γ π))πΆ) β (π΅(π· βΎ (π Γ π))πΆ))) β€ (π΄(π· βΎ (π Γ π))π΅)) |
| 6 | simpr1 1192 | . . . . 5 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 7 | simpr3 1194 | . . . . 5 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β πΆ β π) | |
| 8 | 6, 7 | ovresd 7588 | . . . 4 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)) |
| 9 | simpr2 1193 | . . . . 5 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
| 10 | 9, 7 | ovresd 7588 | . . . 4 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 | 8, 10 | oveq12d 7438 | . . 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) β (π΅(π· βΎ (π Γ π))πΆ)) = ((π΄π·πΆ) β (π΅π·πΆ))) |
| 12 | 11 | fveq2d 6901 | . 2 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (absβ((π΄(π· βΎ (π Γ π))πΆ) β (π΅(π· βΎ (π Γ π))πΆ))) = (absβ((π΄π·πΆ) β (π΅π·πΆ)))) |
| 13 | 6, 9 | ovresd 7588 | . 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΅)) |
| 14 | 5, 12, 13 | 3brtr3d 5179 | 1 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (absβ((π΄π·πΆ) β (π΅π·πΆ))) β€ (π΄π·π΅)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5148 Γ cxp 5676 βΎ cres 5680 βcfv 6548 (class class class)co 7420 β€ cle 11280 β cmin 11475 abscabs 15214 Basecbs 17180 distcds 17242 Metcmet 21265 MetSpcms 24237 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-fz 13518 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-tset 17252 df-ple 17253 df-ds 17255 df-topgen 17425 df-xrs 17484 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-xms 24239 df-ms 24240 |
| This theorem is referenced by: nmrtri 24546 |
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