ballss3 - Mathbox for Glauco Siliprandi

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Theorem ballss3 46381

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 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3		simpr 490	. . . . . . 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 24761	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
8	4, 5, 6, 7	syl3anc 1399	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
9	8	adantr 486	. . . . . . 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 500	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋)
12	10	simprd 501	. . . . 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 1399	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝐴)
15	14	ex 418	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ 𝐴))
16	1, 15	ralrimi 3264	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
17		dfss3 3902	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 Ⅎwnf 1816 ∈ wcel 2147 ∀wral 3080 ⊆ wss 3900 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℝ^*cxr 11381 < clt 11382 PsMetcpsmet 21713 ballcbl 21716

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-map 8887 df-xr 11386 df-psmet 21721 df-bl 21724

This theorem is used by: ioorrnopnlem 47587

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