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 24217

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 24210	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
2	1	ffnd 6657	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋))
3	1	ffvelcdmda 7022	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ ℝ^*)
4		elxp6 7965	. . . . . . . 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 24216	. . . . . . . 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 7970	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = ⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩)
10	9	fveq2d 6830	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = (𝐷‘⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩))
11		df-ov 7356	. . . . . . . 8 ⊢ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)) = (𝐷‘⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩)
12	10, 11	eqtr4di 2782	. . . . . . 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 5123	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷‘𝑎))
15		elxrge0 13378	. . . . 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 3121	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞))
18		fnfvrnss 7059	. . 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 6490	. 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 3044 ⊆ wss 3905 ⟨cop 4585 class class class wbr 5095 × cxp 5621 ran crn 5624 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 2^nd c2nd 7930 0cc0 11028 +∞cpnf 11165 ℝ^*cxr 11167 ≤ cle 11169 [,]cicc 13269 PsMetcpsmet 21263

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 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-po 5531 df-so 5532 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-2 12209 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-icc 13273 df-psmet 21271

This theorem is referenced by: sitmcl 34318

W3C validator