Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > psmetxrge0 | Structured version Visualization version 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 22111 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) | |
2 | ffn 6045 | . . 3 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋)) |
4 | 1 | ffvelrnda 6359 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ ℝ*) |
5 | elxp6 7200 | . . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = 〈(1st ‘𝑎), (2nd ‘𝑎)〉 ∧ ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋))) | |
6 | 5 | simprbi 480 | . . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋)) |
7 | psmetge0 22117 | . . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎))) | |
8 | 7 | 3expb 1266 | . . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1st ‘𝑎) ∈ 𝑋 ∧ (2nd ‘𝑎) ∈ 𝑋)) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎))) |
9 | 6, 8 | sylan2 491 | . . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1st ‘𝑎)𝐷(2nd ‘𝑎))) |
10 | 1st2nd2 7205 | . . . . . . . . 9 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = 〈(1st ‘𝑎), (2nd ‘𝑎)〉) | |
11 | 10 | fveq2d 6195 | . . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = (𝐷‘〈(1st ‘𝑎), (2nd ‘𝑎)〉)) |
12 | df-ov 6653 | . . . . . . . 8 ⊢ ((1st ‘𝑎)𝐷(2nd ‘𝑎)) = (𝐷‘〈(1st ‘𝑎), (2nd ‘𝑎)〉) | |
13 | 11, 12 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = ((1st ‘𝑎)𝐷(2nd ‘𝑎))) |
14 | 13 | adantl 482 | . . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) = ((1st ‘𝑎)𝐷(2nd ‘𝑎))) |
15 | 9, 14 | breqtrrd 4681 | . . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷‘𝑎)) |
16 | elxrge0 12281 | . . . . 5 ⊢ ((𝐷‘𝑎) ∈ (0[,]+∞) ↔ ((𝐷‘𝑎) ∈ ℝ* ∧ 0 ≤ (𝐷‘𝑎))) | |
17 | 4, 15, 16 | sylanbrc 698 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ (0[,]+∞)) |
18 | 17 | ralrimiva 2966 | . . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞)) |
19 | fnfvrnss 6390 | . . 3 ⊢ ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞)) | |
20 | 3, 18, 19 | syl2anc 693 | . 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞)) |
21 | df-f 5892 | . 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞))) | |
22 | 3, 20, 21 | sylanbrc 698 | 1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 〈cop 4183 class class class wbr 4653 × cxp 5112 ran crn 5115 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 [,]cicc 12178 PsMetcpsmet 19730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-psmet 19738 |
This theorem is referenced by: sitmcl 30413 |
Copyright terms: Public domain | W3C validator |