ballss3 - Mathbox for Glauco Siliprandi

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Theorem ballss3 41379

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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3		simpr 487	. . . . . . 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 22997	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
8	4, 5, 6, 7	syl3anc 1367	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
9	8	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
10	3, 9	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))
11	10	simpld 497	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋)
12	10	simprd 498	. . . . 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 1367	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝐴)
15	14	ex 415	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ 𝐴))
16	1, 15	ralrimi 3216	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
17		dfss3 3956	. 2 ⊢ ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
18	16, 17	sylibr 236	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝ^*cxr 10674 < clt 10675 PsMetcpsmet 20529 ballcbl 20532

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 df-xr 10679 df-psmet 20537 df-bl 20540

This theorem is referenced by: ioorrnopnlem 42609

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