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Theorem xmetpsmet 23409

Description: An extended metric is a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)

**Assertion**
Ref	Expression
xmetpsmet	⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))

Proof of Theorem xmetpsmet Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetf 23390	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
2		xmet0 23403	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷𝑥) = 0)
3		3anrot 1098	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋))
4		xmettri2 23401	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
5	3, 4	sylan2br 594	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
6	5	3anassrs 1358	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
7	6	ralrimiva 3107	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
8	7	ralrimiva 3107	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
9	2, 8	jca 511	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
10	9	ralrimiva 3107	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
11		elfvex 6789	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ V)
12		ispsmet 23365	. . 3 ⊢ (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
13	11, 12	syl 17	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ^* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
14	1, 10, 13	mpbir2and 709	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 Vcvv 3422 class class class wbr 5070 × cxp 5578 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 0cc0 10802 ℝ^*cxr 10939 ≤ cle 10941 +_𝑒 cxad 12775 PsMetcpsmet 20494 ∞Metcxmet 20495

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-xr 10944 df-psmet 20502 df-xmet 20503

This theorem is referenced by: blfval 23445 xmetutop 23630 xmsusp 23631 cfilucfil3 24389 cmetcusp 24423 cnflduss 24425 reust 24450 qqhucn 31842 sitmcl 32218 heicant 35739 metpsmet 42530 ioorrnopnlem 43735

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