ballss3 - Mathbox for Glauco Siliprandi

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Theorem ballss3 42165

Description: A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
ballss3.y	⊢ Ⅎ𝑥𝜑
ballss3.d	⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑋))
ballss3.p	⊢ (𝜑 → 𝑃 ∈ 𝑋)
ballss3.r	⊢ (𝜑 → 𝑅 ∈ ℝ^*)
ballss3.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥 ∈ 𝐴)

**Assertion**
Ref	Expression
ballss3	⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐷 𝑥,𝑃 𝑥,𝑅 Allowed substitution hints: 𝜑(𝑥) 𝑋(𝑥)

**Proof of Theorem ballss3**
Step	Hyp	Ref	Expression
1		ballss3.y	. . 3 ⊢ Ⅎ𝑥𝜑
2		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ (𝑃(ball‘𝐷)𝑅))
4		ballss3.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑋))
5		ballss3.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ 𝑋)
6		ballss3.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
7		elblps 23133	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
8	4, 5, 6, 7	syl3anc 1372	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
9	8	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
10	3, 9	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))
11	10	simpld 498	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋)
12	10	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝑥) < 𝑅)
13		ballss3.a	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥 ∈ 𝐴)
14	2, 11, 12, 13	syl3anc 1372	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝐴)
15	14	ex 416	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ 𝐴))
16	1, 15	ralrimi 3127	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
17		dfss3 3863	. 2 ⊢ ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
18	16, 17	sylibr 237	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 Ⅎwnf 1790 ∈ wcel 2113 ∀wral 3053 ⊆ wss 3841 class class class wbr 5027 ‘cfv 6333 (class class class)co 7164 ℝ^*cxr 10745 < clt 10746 PsMetcpsmet 20194 ballcbl 20197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-map 8432 df-xr 10750 df-psmet 20202 df-bl 20205

This theorem is referenced by: ioorrnopnlem 43371

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