Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ballss3 | Structured version Visualization version GIF version |
Description: A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
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) |
Ref | Expression |
---|---|
ballss3 | ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴) |
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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