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Theorem psmetxrge0 22926
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 22919 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21ffnd 6505 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋))
31ffvelrnda 6843 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) ∈ ℝ*)
4 elxp6 7719 . . . . . . . 8 (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = ⟨(1st𝑎), (2nd𝑎)⟩ ∧ ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋)))
54simprbi 500 . . . . . . 7 (𝑎 ∈ (𝑋 × 𝑋) → ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋))
6 psmetge0 22925 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
763expb 1117 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋)) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
85, 7sylan2 595 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
9 1st2nd2 7724 . . . . . . . . 9 (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = ⟨(1st𝑎), (2nd𝑎)⟩)
109fveq2d 6666 . . . . . . . 8 (𝑎 ∈ (𝑋 × 𝑋) → (𝐷𝑎) = (𝐷‘⟨(1st𝑎), (2nd𝑎)⟩))
11 df-ov 7153 . . . . . . . 8 ((1st𝑎)𝐷(2nd𝑎)) = (𝐷‘⟨(1st𝑎), (2nd𝑎)⟩)
1210, 11syl6eqr 2877 . . . . . . 7 (𝑎 ∈ (𝑋 × 𝑋) → (𝐷𝑎) = ((1st𝑎)𝐷(2nd𝑎)))
1312adantl 485 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) = ((1st𝑎)𝐷(2nd𝑎)))
148, 13breqtrrd 5081 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷𝑎))
15 elxrge0 12847 . . . . 5 ((𝐷𝑎) ∈ (0[,]+∞) ↔ ((𝐷𝑎) ∈ ℝ* ∧ 0 ≤ (𝐷𝑎)))
163, 14, 15sylanbrc 586 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) ∈ (0[,]+∞))
1716ralrimiva 3177 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷𝑎) ∈ (0[,]+∞))
18 fnfvrnss 6876 . . 3 ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞))
192, 17, 18syl2anc 587 . 2 (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞))
20 df-f 6348 . 2 (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞)))
212, 19, 20sylanbrc 586 1 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3920  cop 4557   class class class wbr 5053   × cxp 5541  ran crn 5544   Fn wfn 6339  wf 6340  cfv 6344  (class class class)co 7150  1st c1st 7683  2nd c2nd 7684  0cc0 10536  +∞cpnf 10671  *cxr 10673  cle 10675  [,]cicc 12741  PsMetcpsmet 20532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-po 5462  df-so 5463  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-er 8286  df-map 8405  df-en 8507  df-dom 8508  df-sdom 8509  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-2 11700  df-rp 12390  df-xneg 12507  df-xadd 12508  df-xmul 12509  df-icc 12745  df-psmet 20540
This theorem is referenced by:  sitmcl  31669
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