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Mirrors > Home > ILE Home > Th. List > mettri2 | GIF version |
Description: Triangle inequality for the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
mettri2 | ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metxmet 12344 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | xmettri2 12350 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) | |
3 | 1, 2 | sylan 279 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) |
4 | metcl 12342 | . . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶𝐷𝐴) ∈ ℝ) | |
5 | 4 | 3adant3r3 1175 | . . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐶𝐷𝐴) ∈ ℝ) |
6 | metcl 12342 | . . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐶𝐷𝐵) ∈ ℝ) | |
7 | 6 | 3adant3r2 1174 | . . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐶𝐷𝐵) ∈ ℝ) |
8 | rexadd 9528 | . . 3 ⊢ (((𝐶𝐷𝐴) ∈ ℝ ∧ (𝐶𝐷𝐵) ∈ ℝ) → ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)) = ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) | |
9 | 5, 7, 8 | syl2anc 406 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)) = ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) |
10 | 3, 9 | breqtrd 3919 | 1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 945 = wceq 1314 ∈ wcel 1463 class class class wbr 3895 ‘cfv 5081 (class class class)co 5728 ℝcr 7546 + caddc 7550 ≤ cle 7725 +𝑒 cxad 9450 ∞Metcxmet 11992 Metcmet 11993 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1re 7639 ax-addrcl 7642 ax-rnegex 7654 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-if 3441 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-map 6498 df-pnf 7726 df-mnf 7727 df-xr 7728 df-xadd 9453 df-xmet 12000 df-met 12001 |
This theorem is referenced by: mettri 12362 mstri2 12460 |
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