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Theorem psmetxrge0 22099
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 22092 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2 ffn 6032 . . 3 (𝐷:(𝑋 × 𝑋)⟶ℝ*𝐷 Fn (𝑋 × 𝑋))
31, 2syl 17 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋))
41ffvelrnda 6345 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) ∈ ℝ*)
5 elxp6 7185 . . . . . . . 8 (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = ⟨(1st𝑎), (2nd𝑎)⟩ ∧ ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋)))
65simprbi 480 . . . . . . 7 (𝑎 ∈ (𝑋 × 𝑋) → ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋))
7 psmetge0 22098 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
873expb 1264 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1st𝑎) ∈ 𝑋 ∧ (2nd𝑎) ∈ 𝑋)) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
96, 8sylan2 491 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1st𝑎)𝐷(2nd𝑎)))
10 1st2nd2 7190 . . . . . . . . 9 (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = ⟨(1st𝑎), (2nd𝑎)⟩)
1110fveq2d 6182 . . . . . . . 8 (𝑎 ∈ (𝑋 × 𝑋) → (𝐷𝑎) = (𝐷‘⟨(1st𝑎), (2nd𝑎)⟩))
12 df-ov 6638 . . . . . . . 8 ((1st𝑎)𝐷(2nd𝑎)) = (𝐷‘⟨(1st𝑎), (2nd𝑎)⟩)
1311, 12syl6eqr 2672 . . . . . . 7 (𝑎 ∈ (𝑋 × 𝑋) → (𝐷𝑎) = ((1st𝑎)𝐷(2nd𝑎)))
1413adantl 482 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) = ((1st𝑎)𝐷(2nd𝑎)))
159, 14breqtrrd 4672 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷𝑎))
16 elxrge0 12266 . . . . 5 ((𝐷𝑎) ∈ (0[,]+∞) ↔ ((𝐷𝑎) ∈ ℝ* ∧ 0 ≤ (𝐷𝑎)))
174, 15, 16sylanbrc 697 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷𝑎) ∈ (0[,]+∞))
1817ralrimiva 2963 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷𝑎) ∈ (0[,]+∞))
19 fnfvrnss 6376 . . 3 ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞))
203, 18, 19syl2anc 692 . 2 (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞))
21 df-f 5880 . 2 (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞)))
223, 20, 21sylanbrc 697 1 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  wss 3567  cop 4174   class class class wbr 4644   × cxp 5102  ran crn 5105   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  0cc0 9921  +∞cpnf 10056  *cxr 10058  cle 10060  [,]cicc 12163  PsMetcpsmet 19711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-po 5025  df-so 5026  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-2 11064  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-icc 12167  df-psmet 19719
This theorem is referenced by:  sitmcl  30387
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