Theorem psmetxrge0 14652

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.

Step	Hyp	Ref	Expression
1		psmetf 14645	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2	1	ffnd 5411	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋))
3	1	ffvelcdmda 5700	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ ℝ*)
4		elxp6 6236	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = ⟨(1st ‘𝑎), (2nd ‘𝑎)⟩ ∧ ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋)))
5	4	simprbi 275	. . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋))
6		psmetge0 14651	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎)))
7	6	3expb 1206	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋)) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎)))
8	5, 7	sylan2 286	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎)))
9		1st2nd2 6242	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = ⟨(1st ‘𝑎), (2nd ‘𝑎)⟩)
10	9	fveq2d 5565	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = (𝐷‘⟨(1st ‘𝑎), (2nd ‘𝑎)⟩))
11		df-ov 5928	. . . . . . . 8 ⊢ ((1st ‘𝑎)𝐷(2nd ‘𝑎)) = (𝐷‘⟨(1st ‘𝑎), (2nd ‘𝑎)⟩)
12	10, 11	eqtr4di 2247	. . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = ((1st ‘𝑎)𝐷(2nd ‘𝑎)))
13	12	adantl 277	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) = ((1st ‘𝑎)𝐷(2nd ‘𝑎)))
14	8, 13	breqtrrd 4062	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷‘𝑎))
15		elxrge0 10070	. . . . 5 ⊢ ((𝐷‘𝑎) ∈ (0[,]+∞) ↔ ((𝐷‘𝑎) ∈ ℝ* ∧ 0 ≤ (𝐷‘𝑎)))
16	3, 14, 15	sylanbrc 417	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ (0[,]+∞))
17	16	ralrimiva 2570	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞))
18		fnfvrnss 5725	. . 3 ⊢ ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞))
19	2, 17, 18	syl2anc 411	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞))
20		df-f 5263	. 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞)))
21	2, 19, 20	sylanbrc 417	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  ⟨cop 3626   class class class wbr 4034   × cxp 4662  ran crn 4665   Fn wfn 5254  ⟶wf 5255  ‘cfv 5259  (class class class)co 5925  1^st c1st 6205  2^nd c2nd 6206  0cc0 7896  +∞cpnf 8075  ℝ^*cxr 8077   ≤ cle 8079  [,]cicc 9983  PsMetcpsmet 14167

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012  ax-pre-mulgt0 8013

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-2 9066  df-xadd 9865  df-icc 9987  df-psmet 14175

This theorem is referenced by: (None)