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Theorem psmetxrge0 24288

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 24281	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
2	1	ffnd 6663	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 Fn (𝑋 × 𝑋))
3	1	ffvelcdmda 7030	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ ℝ^*)
4		elxp6 7969	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) ↔ (𝑎 = ⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩ ∧ ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋)))
5	4	simprbi 497	. . . . . . 7 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋))
6		psmetge0 24287	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
7	6	3expb 1121	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ((1^st ‘𝑎) ∈ 𝑋 ∧ (2^nd ‘𝑎) ∈ 𝑋)) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
8	5, 7	sylan2 594	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)))
9		1st2nd2 7974	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → 𝑎 = ⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩)
10	9	fveq2d 6838	. . . . . . . 8 ⊢ (𝑎 ∈ (𝑋 × 𝑋) → (𝐷‘𝑎) = (𝐷‘⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩))
11		df-ov 7363	. . . . . . . 8 ⊢ ((1^st ‘𝑎)𝐷(2^nd ‘𝑎)) = (𝐷‘⟨(1^st ‘𝑎), (2^nd ‘𝑎)⟩)
12	10, 11	eqtr4di 2790	. . . . . . 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 5114	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → 0 ≤ (𝐷‘𝑎))
15		elxrge0 13401	. . . . 5 ⊢ ((𝐷‘𝑎) ∈ (0[,]+∞) ↔ ((𝐷‘𝑎) ∈ ℝ^* ∧ 0 ≤ (𝐷‘𝑎)))
16	3, 14, 15	sylanbrc 584	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ (𝑋 × 𝑋)) → (𝐷‘𝑎) ∈ (0[,]+∞))
17	16	ralrimiva 3130	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞))
18		fnfvrnss 7067	. . 3 ⊢ ((𝐷 Fn (𝑋 × 𝑋) ∧ ∀𝑎 ∈ (𝑋 × 𝑋)(𝐷‘𝑎) ∈ (0[,]+∞)) → ran 𝐷 ⊆ (0[,]+∞))
19	2, 17, 18	syl2anc 585	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ran 𝐷 ⊆ (0[,]+∞))
20		df-f 6496	. 2 ⊢ (𝐷:(𝑋 × 𝑋)⟶(0[,]+∞) ↔ (𝐷 Fn (𝑋 × 𝑋) ∧ ran 𝐷 ⊆ (0[,]+∞)))
21	2, 19, 20	sylanbrc 584	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3890 ⟨cop 4574 class class class wbr 5086 × cxp 5622 ran crn 5625 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 0cc0 11029 +∞cpnf 11167 ℝ^*cxr 11169 ≤ cle 11171 [,]cicc 13292 PsMetcpsmet 21328

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-2 12235 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-icc 13296 df-psmet 21336

This theorem is referenced by: sitmcl 34511

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