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| Mirrors > Home > MPE Home > Th. List > psmetge0 | Structured version Visualization version GIF version | ||
| Description: The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| Ref | Expression |
|---|---|
| psmetge0 | ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1137 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | |
| 2 | simp2 1138 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
| 3 | simp3 1139 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
| 4 | psmettri2 24319 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐷𝐵) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵))) | |
| 5 | 1, 2, 3, 3, 4 | syl13anc 1374 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵))) |
| 6 | 2re 12340 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 7 | rexr 11307 | . . . . 5 ⊢ (2 ∈ ℝ → 2 ∈ ℝ*) | |
| 8 | xmul01 13309 | . . . . 5 ⊢ (2 ∈ ℝ* → (2 ·e 0) = 0) | |
| 9 | 6, 7, 8 | mp2b 10 | . . . 4 ⊢ (2 ·e 0) = 0 |
| 10 | psmet0 24318 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 0) | |
| 11 | 10 | 3adant2 1132 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 0) |
| 12 | 9, 11 | eqtr4id 2796 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 ·e 0) = (𝐵𝐷𝐵)) |
| 13 | psmetcl 24317 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
| 14 | x2times 13341 | . . . 4 ⊢ ((𝐴𝐷𝐵) ∈ ℝ* → (2 ·e (𝐴𝐷𝐵)) = ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵))) | |
| 15 | 13, 14 | syl 17 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 ·e (𝐴𝐷𝐵)) = ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵))) |
| 16 | 5, 12, 15 | 3brtr4d 5175 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (2 ·e 0) ≤ (2 ·e (𝐴𝐷𝐵))) |
| 17 | 0xr 11308 | . . 3 ⊢ 0 ∈ ℝ* | |
| 18 | 2rp 13039 | . . . 4 ⊢ 2 ∈ ℝ+ | |
| 19 | 18 | a1i 11 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 2 ∈ ℝ+) |
| 20 | xlemul2 13333 | . . 3 ⊢ ((0 ∈ ℝ* ∧ (𝐴𝐷𝐵) ∈ ℝ* ∧ 2 ∈ ℝ+) → (0 ≤ (𝐴𝐷𝐵) ↔ (2 ·e 0) ≤ (2 ·e (𝐴𝐷𝐵)))) | |
| 21 | 17, 13, 19, 20 | mp3an2i 1468 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (0 ≤ (𝐴𝐷𝐵) ↔ (2 ·e 0) ≤ (2 ·e (𝐴𝐷𝐵)))) |
| 22 | 16, 21 | mpbird 257 | 1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 ℝ*cxr 11294 ≤ cle 11296 2c2 12321 ℝ+crp 13034 +𝑒 cxad 13152 ·e cxmu 13153 PsMetcpsmet 21348 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-psmet 21356 |
| This theorem is referenced by: psmetxrge0 24323 psmetlecl 24325 distspace 24326 xblpnfps 24405 xblss2ps 24411 metustexhalf 24569 blval2 24575 metuel2 24578 metider 33893 |
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