psmetxrge0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > psmetxrge0		Structured version Visualization version GIF version

Theorem psmetxrge0 24257

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 24250	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
2	1	ffnd 6712	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋))
3	1	ffvelcdmda 7079	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ ℝ^*)
4		elxp6 8027	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = ⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩ ∧ ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋)))
5	4	simprbi 496	. . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋))
6		psmetge0 24256	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
7	6	3expb 1120	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋)) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
8	5, 7	sylan2 593	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
9		1st2nd2 8032	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = ⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩)
10	9	fveq2d 6885	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = (𝐷‘⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩))
11		df-ov 7413	. . . . . . . 8 ⊢ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)) = (𝐷‘⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩)
12	10, 11	eqtr4di 2789	. . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
13	12	adantl 481	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) = ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
14	8, 13	breqtrrd 5152	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷‘𝑎))
15		elxrge0 13479	. . . . 5 ⊢ ((𝐷‘𝑎) ∈ (0[,]+∞) ↔ ((𝐷‘𝑎) ∈ ℝ^* ∧ 0 ≤ (𝐷‘𝑎)))
16	3, 14, 15	sylanbrc 583	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ (0[,]+∞))
17	16	ralrimiva 3133	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞))
18		fnfvrnss 7116	. . 3 ⊢ ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞))
19	2, 17, 18	syl2anc 584	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞))
20		df-f 6540	. 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞)))
21	2, 19, 20	sylanbrc 583	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3052 ⊆ wss 3931 ⟨cop 4612 class class class wbr 5124 × cxp 5657 ran crn 5660 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 0cc0 11134 +∞cpnf 11271 ℝ^*cxr 11273 ≤ cle 11275 [,]cicc 13370 PsMetcpsmet 21304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-po 5566 df-so 5567 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7993 df-2nd 7994 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-2 12308 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-icc 13374 df-psmet 21312

This theorem is referenced by: sitmcl 34388

W3C validator