Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > msrtri | 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 14348 | . . 3 β’ (π β MetSp β (π· βΎ (π Γ π)) β (Metβπ)) |
| 4 | metrtri 14261 | . . 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 283 | . 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 1005 | . . . . 5 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 7 | simpr3 1007 | . . . . 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 6032 | . . . 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 1006 | . . . . 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 6032 | . . . 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 5909 | . . 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 5534 | . 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 6032 | . 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 4049 | 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 set class |
| Syntax hints: β wi 4 β§ wa 104 β§ w3a 980 = wceq 1364 β wcel 2160 class class class wbr 4018 Γ cxp 4639 βΎ cres 4643 βcfv 5231 (class class class)co 5891 β€ cle 8011 β cmin 8146 abscabs 11024 Basecbs 12480 distcds 12564 Metcmet 13811 MetSpcms 14221 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 ax-arch 7948 ax-caucvg 7949 |
| This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-map 6668 df-sup 7001 df-inf 7002 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 df-7 9001 df-8 9002 df-9 9003 df-n0 9195 df-z 9272 df-uz 9547 df-q 9638 df-rp 9672 df-xneg 9790 df-xadd 9791 df-seqfrec 10464 df-exp 10538 df-cj 10869 df-re 10870 df-im 10871 df-rsqrt 11025 df-abs 11026 df-ndx 12483 df-slot 12484 df-base 12486 df-tset 12574 df-rest 12712 df-topn 12713 df-topgen 12731 df-psmet 13817 df-xmet 13818 df-met 13819 df-bl 13820 df-mopn 13821 df-top 13882 df-topon 13895 df-topsp 13915 df-bases 13927 df-xms 14223 df-ms 14224 |
| This theorem is referenced by: (None) |
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