Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mettri2 | Structured version Visualization version 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 22871 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | xmettri2 22877 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) | |
3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) |
4 | metcl 22869 | . . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶𝐷𝐴) ∈ ℝ) | |
5 | 4 | 3adant3r3 1176 | . . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐶𝐷𝐴) ∈ ℝ) |
6 | metcl 22869 | . . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐶𝐷𝐵) ∈ ℝ) | |
7 | 6 | 3adant3r2 1175 | . . 3 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐶𝐷𝐵) ∈ ℝ) |
8 | 5, 7 | rexaddd 12615 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)) = ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) |
9 | 3, 8 | breqtrd 5083 | 1 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 + caddc 10528 ≤ cle 10664 +𝑒 cxad 12493 ∞Metcxmet 20458 Metcmet 20459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-i2m1 10593 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-xadd 12496 df-xmet 20466 df-met 20467 |
This theorem is referenced by: mettri 22889 mstri2 23004 metf1o 34911 isbnd3 34943 heibor1lem 34968 bfplem2 34982 |
Copyright terms: Public domain | W3C validator |