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Theorem xmetpsmet 24340

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 24321	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
2		xmet0 24334	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐷𝑥) = 0)
3		3anrot 1097	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋))
4		xmettri2 24332	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
5	3, 4	sylan2br 593	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
6	5	3anassrs 1357	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
7	6	ralrimiva 3136	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
8	7	ralrimiva 3136	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))
9	2, 8	jca 510	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
10	9	ralrimiva 3136	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
11		elfvex 6929	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ V)
12		ispsmet 24296	. . 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 711	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3463 class class class wbr 5144 × cxp 5671 ⟶wf 6540 ‘cfv 6544 (class class class)co 7414 0cc0 11147 ℝ^*cxr 11286 ≤ cle 11288 +_𝑒 cxad 13136 PsMetcpsmet 21321 ∞Metcxmet 21322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8847 df-xr 11291 df-psmet 21329 df-xmet 21330

This theorem is referenced by: blfval 24376 xmetutop 24563 xmsusp 24564 cfilucfil3 25334 cmetcusp 25368 cnflduss 25370 reust 25395 qqhucn 33818 sitmcl 34196 heicant 37367 metpsmet 44727 ioorrnopnlem 45959

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