elbl2ps - Metamath Proof Explorer

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Theorem elbl2ps 24299

Description: Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.)

**Assertion**
Ref	Expression
elbl2ps	⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))

**Proof of Theorem elbl2ps**
Step	Hyp	Ref	Expression
1		simprr 772	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
2		elblps 24297	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
3	2	3expa 1118	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑅 ∈ ℝ^*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
4	3	an32s 652	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
5	4	adantrr 717	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
6	1, 5	mpbirand 707	1 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2111 class class class wbr 5086 ‘cfv 6476 (class class class)co 7341 ℝ^*cxr 11140 < clt 11141 PsMetcpsmet 21270 ballcbl 21273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5506 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-fv 6484 df-ov 7344 df-oprab 7345 df-mpo 7346 df-1st 7916 df-2nd 7917 df-map 8747 df-xr 11145 df-psmet 21278 df-bl 21281

This theorem is referenced by: elbl3ps 24301 blcomps 24303

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