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Mirrors > Home > ILE Home > Th. List > psmetxrge0 | GIF version |
Description: The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
Ref | Expression |
---|---|
psmetxrge0 | ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psmetf 12494 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | 1 | ffnd 5273 | . 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋)) |
3 | 1 | ffvelrnda 5555 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ ℝ*) |
4 | elxp6 6067 | . . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = 〈(1st ‘𝑎), (2nd ‘𝑎)〉 ∧ ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋))) | |
5 | 4 | simprbi 273 | . . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋)) |
6 | psmetge0 12500 | . . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎))) | |
7 | 6 | 3expb 1182 | . . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋)) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎))) |
8 | 5, 7 | sylan2 284 | . . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎))) |
9 | 1st2nd2 6073 | . . . . . . . . 9 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = 〈(1st ‘𝑎), (2nd ‘𝑎)〉) | |
10 | 9 | fveq2d 5425 | . . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = (𝐷‘〈(1st ‘𝑎), (2nd ‘𝑎)〉)) |
11 | df-ov 5777 | . . . . . . . 8 ⊢ ((1st ‘𝑎)𝐷(2nd ‘𝑎)) = (𝐷‘〈(1st ‘𝑎), (2nd ‘𝑎)〉) | |
12 | 10, 11 | syl6eqr 2190 | . . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = ((1st ‘𝑎)𝐷(2nd ‘𝑎))) |
13 | 12 | adantl 275 | . . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) = ((1st ‘𝑎)𝐷(2nd ‘𝑎))) |
14 | 8, 13 | breqtrrd 3956 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷‘𝑎)) |
15 | elxrge0 9761 | . . . . 5 ⊢ ((𝐷‘𝑎) ∈ (0[,]+∞) ↔ ((𝐷‘𝑎) ∈ ℝ* ∧ 0 ≤ (𝐷‘𝑎))) | |
16 | 3, 14, 15 | sylanbrc 413 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ (0[,]+∞)) |
17 | 16 | ralrimiva 2505 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞)) |
18 | fnfvrnss 5580 | . . 3 ⊢ ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞)) | |
19 | 2, 17, 18 | syl2anc 408 | . 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞)) |
20 | df-f 5127 | . 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞))) | |
21 | 2, 19, 20 | sylanbrc 413 | 1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 〈cop 3530 class class class wbr 3929 × cxp 4537 ran crn 4540 Fn wfn 5118 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 1st c1st 6036 2nd c2nd 6037 0cc0 7620 +∞cpnf 7797 ℝ*cxr 7799 ≤ cle 7801 [,]cicc 9674 PsMetcpsmet 12148 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-2 8779 df-xadd 9560 df-icc 9678 df-psmet 12156 |
This theorem is referenced by: (None) |
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