psmetxrge0 - Metamath Proof Explorer

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Theorem psmetxrge0 24638

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 24631	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
2	1	ffnd 6711	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋))
3	1	ffvelcdmda 7085	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ ℝ^*)
4		elxp6 8036	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = ⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩ ∧ ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋)))
5	4	simprbi 503	. . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋))
6		psmetge0 24637	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
7	6	3expb 1138	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋)) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
8	5, 7	sylan2 605	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
9		1st2nd2 8041	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = ⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩)
10	9	fveq2d 6890	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = (𝐷‘⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩))
11		df-ov 7424	. . . . . . . 8 ⊢ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)) = (𝐷‘⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩)
12	10, 11	eqtr4di 2813	. . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
13	12	adantl 487	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) = ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
14	8, 13	breqtrrd 5132	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷‘𝑎))
15		elxrge0 13592	. . . . 5 ⊢ ((𝐷‘𝑎) ∈ (0[,]+∞) ↔ ((𝐷‘𝑎) ∈ ℝ^* ∧ 0 ≤ (𝐷‘𝑎)))
16	3, 14, 15	sylanbrc 595	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ (0[,]+∞))
17	16	ralrimiva 3154	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞))
18		fnfvrnss 7122	. . 3 ⊢ ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞))
19	2, 17, 18	syl2anc 596	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞))
20		df-f 6543	. 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞)))
21	2, 19, 20	sylanbrc 595	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ⊆ wss 3895 ⟨cop 4834 class class class wbr 5102 × cxp 5648 ran crn 5651 Fn wfn 6534 ⟶wf 6535 ‘cfv 6539 (class class class)co 7421 1^st c1st 8000 2^nd c2nd 8001 0cc0 11204 +∞cpnf 11344 ℝ^*cxr 11346 ≤ cle 11348 [,]cicc 13483 PsMetcpsmet 21668

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-er 8721 df-map 8853 df-en 8978 df-dom 8979 df-sdom 8980 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-icc 13487 df-psmet 21676

This theorem is used by: sitmcl 35030

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