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 14191 | . . 3 β’ (π β MetSp β (π· βΎ (π Γ π)) β (Metβπ)) |
| 4 | metrtri 14104 | . . 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 1004 | . . . . 5 β’ ((π β MetSp β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β π΄ β π) | |
| 7 | simpr3 1006 | . . . . 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 6028 | . . . 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 1005 | . . . . 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 6028 | . . . 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 5906 | . . 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 5531 | . 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 6028 | . 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 4046 | 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 979 = wceq 1363 β wcel 2158 class class class wbr 4015 Γ cxp 4636 βΎ cres 4640 βcfv 5228 (class class class)co 5888 β€ cle 8006 β cmin 8141 abscabs 11019 Basecbs 12475 distcds 12559 Metcmet 13654 MetSpcms 14064 |
| 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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944 |
| This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-map 6663 df-sup 6996 df-inf 6997 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-7 8996 df-8 8997 df-9 8998 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-xneg 9785 df-xadd 9786 df-seqfrec 10459 df-exp 10533 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-ndx 12478 df-slot 12479 df-base 12481 df-tset 12569 df-rest 12707 df-topn 12708 df-topgen 12726 df-psmet 13660 df-xmet 13661 df-met 13662 df-bl 13663 df-mopn 13664 df-top 13725 df-topon 13738 df-topsp 13758 df-bases 13770 df-xms 14066 df-ms 14067 |
| This theorem is referenced by: (None) |
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