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Mirrors > Home > ILE Home > Th. List > psmetge0 | GIF version |
Description: The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.) |
Ref | Expression |
---|---|
psmetge0 | ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7981 | . . . 4 ⊢ 0 ∈ ℝ* | |
2 | xaddid1 9836 | . . . 4 ⊢ (0 ∈ ℝ* → (0 +𝑒 0) = 0) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (0 +𝑒 0) = 0 |
4 | psmet0 13460 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 0) | |
5 | 4 | 3adant2 1016 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 0) |
6 | simp1 997 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐷 ∈ (PsMet‘𝑋)) | |
7 | simp2 998 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
8 | simp3 999 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
9 | psmettri2 13461 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵𝐷𝐵) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵))) | |
10 | 6, 7, 8, 8, 9 | syl13anc 1240 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵))) |
11 | 5, 10 | eqbrtrrd 4024 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵))) |
12 | 3, 11 | eqbrtrid 4035 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (0 +𝑒 0) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵))) |
13 | psmetcl 13459 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
14 | xleaddadd 9861 | . . 3 ⊢ ((0 ∈ ℝ* ∧ (𝐴𝐷𝐵) ∈ ℝ*) → (0 ≤ (𝐴𝐷𝐵) ↔ (0 +𝑒 0) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵)))) | |
15 | 1, 13, 14 | sylancr 414 | . 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (0 ≤ (𝐴𝐷𝐵) ↔ (0 +𝑒 0) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐴𝐷𝐵)))) |
16 | 12, 15 | mpbird 167 | 1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 0cc0 7789 ℝ*cxr 7968 ≤ cle 7970 +𝑒 cxad 9744 PsMetcpsmet 13112 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-lttrn 7903 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-map 6643 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-2 8954 df-xadd 9747 df-psmet 13120 |
This theorem is referenced by: psmetxrge0 13465 psmetlecl 13467 distspace 13468 xblpnfps 13531 xblss2ps 13537 |
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