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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 | ⊢ 𝑋 = (Base‘𝑀) |
mscl.d | ⊢ 𝐷 = (dist‘𝑀) |
Ref | Expression |
---|---|
msrtri | ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mscl.x | . . . 4 ⊢ 𝑋 = (Base‘𝑀) | |
2 | mscl.d | . . . 4 ⊢ 𝐷 = (dist‘𝑀) | |
3 | 1, 2 | msmet2 13011 | . . 3 ⊢ (𝑀 ∈ MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (Met‘𝑋)) |
4 | metrtri 12924 | . . 3 ⊢ (((𝐷 ↾ (𝑋 × 𝑋)) ∈ (Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐶) − (𝐵(𝐷 ↾ (𝑋 × 𝑋))𝐶))) ≤ (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵)) | |
5 | 3, 4 | sylan 281 | . 2 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐶) − (𝐵(𝐷 ↾ (𝑋 × 𝑋))𝐶))) ≤ (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵)) |
6 | simpr1 992 | . . . . 5 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
7 | simpr3 994 | . . . . 5 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋) | |
8 | 6, 7 | ovresd 5973 | . . . 4 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐶) = (𝐴𝐷𝐶)) |
9 | simpr2 993 | . . . . 5 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
10 | 9, 7 | ovresd 5973 | . . . 4 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵(𝐷 ↾ (𝑋 × 𝑋))𝐶) = (𝐵𝐷𝐶)) |
11 | 8, 10 | oveq12d 5854 | . . 3 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐶) − (𝐵(𝐷 ↾ (𝑋 × 𝑋))𝐶)) = ((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) |
12 | 11 | fveq2d 5484 | . 2 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐶) − (𝐵(𝐷 ↾ (𝑋 × 𝑋))𝐶))) = (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶)))) |
13 | 6, 9 | ovresd 5973 | . 2 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
14 | 5, 12, 13 | 3brtr3d 4007 | 1 ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 × cxp 4596 ↾ cres 4600 ‘cfv 5182 (class class class)co 5836 ≤ cle 7925 − cmin 8060 abscabs 10925 Basecbs 12337 distcds 12408 Metcmet 12528 MetSpcms 12884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-map 6607 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-xneg 9699 df-xadd 9700 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-ndx 12340 df-slot 12341 df-base 12343 df-tset 12418 df-rest 12500 df-topn 12501 df-topgen 12519 df-psmet 12534 df-xmet 12535 df-met 12536 df-bl 12537 df-mopn 12538 df-top 12543 df-topon 12556 df-topsp 12576 df-bases 12588 df-xms 12886 df-ms 12887 |
This theorem is referenced by: (None) |
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