ballss3 - Mathbox for Glauco Siliprandi

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Theorem ballss3 45523

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 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑)
3		simpr 484	. . . . . . 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 24352	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
8	4, 5, 6, 7	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
10	3, 9	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))
11	10	simpld 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋)
12	10	simprd 495	. . . . 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 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝐴)
15	14	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ 𝐴))
16	1, 15	ralrimi 3235	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
17		dfss3 3910	. 2 ⊢ ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴)
18	16, 17	sylibr 234	1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 Ⅎwnf 1785 ∈ wcel 2114 ∀wral 3051 ⊆ wss 3889 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℝ^*cxr 11178 < clt 11179 PsMetcpsmet 21336 ballcbl 21339

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-map 8775 df-xr 11183 df-psmet 21344 df-bl 21347

This theorem is referenced by: ioorrnopnlem 46732

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