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Theorem psmetxrge0 12515
Description: The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
Assertion
Ref Expression
psmetxrge0 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))

Proof of Theorem psmetxrge0
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 psmetf 12508 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21ffnd 5273 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋))
31ffvelrnda 5555 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) ∈ ℝ*)
4 elxp6 6067 . . . . . . . 8 (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = ⟨(1st𝑎), (2nd𝑎)⟩ ∧ ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋)))
54simprbi 273 . . . . . . 7 (𝑎 ∈ (𝑋 × 𝑋) → ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋))
6 psmetge0 12514 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
763expb 1182 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋)) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
85, 7sylan2 284 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
9 1st2nd2 6073 . . . . . . . . 9 (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = ⟨(1st𝑎), (2nd𝑎)⟩)
109fveq2d 5425 . . . . . . . 8 (𝑎 ∈ (𝑋 × 𝑋) → (𝐷𝑎) = (𝐷‘⟨(1st𝑎), (2nd𝑎)⟩))
11 df-ov 5777 . . . . . . . 8 ((1st𝑎)𝐷(2nd𝑎)) = (𝐷‘⟨(1st𝑎), (2nd𝑎)⟩)
1210, 11syl6eqr 2190 . . . . . . 7 (𝑎 ∈ (𝑋 × 𝑋) → (𝐷𝑎) = ((1st𝑎)𝐷(2nd𝑎)))
1312adantl 275 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) = ((1st𝑎)𝐷(2nd𝑎)))
148, 13breqtrrd 3956 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷𝑎))
15 elxrge0 9773 . . . . 5 ((𝐷𝑎) ∈ (0[,]+∞) ↔ ((𝐷𝑎) ∈ ℝ* ∧ 0 ≤ (𝐷𝑎)))
163, 14, 15sylanbrc 413 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) ∈ (0[,]+∞))
1716ralrimiva 2505 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷𝑎) ∈ (0[,]+∞))
18 fnfvrnss 5580 . . 3 ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞))
192, 17, 18syl2anc 408 . 2 (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞))
20 df-f 5127 . 2 (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞)))
212, 19, 20sylanbrc 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 7632  +∞cpnf 7809  *cxr 7811  cle 7813  [,]cicc 9686  PsMetcpsmet 12162
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 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-lttrn 7746  ax-pre-ltadd 7748  ax-pre-mulgt0 7749
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 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-2 8791  df-xadd 9572  df-icc 9690  df-psmet 12170
This theorem is referenced by: (None)
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