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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 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑) | |
3 | simpr 479 | . . . . . . 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 22611 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))) | |
8 | 4, 5, 6, 7 | syl3anc 1439 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))) |
9 | 8 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))) |
10 | 3, 9 | mpbid 224 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)) |
11 | 10 | simpld 490 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋) |
12 | 10 | simprd 491 | . . . . 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 1439 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝐴) |
15 | 14 | ex 403 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ 𝐴)) |
16 | 1, 15 | ralrimi 3139 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴) |
17 | dfss3 3810 | . 2 ⊢ ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴) | |
18 | 16, 17 | sylibr 226 | 1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 Ⅎwnf 1827 ∈ wcel 2107 ∀wral 3090 ⊆ wss 3792 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℝ*cxr 10412 < clt 10413 PsMetcpsmet 20137 ballcbl 20140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-map 8144 df-xr 10417 df-psmet 20145 df-bl 20148 |
This theorem is referenced by: ioorrnopnlem 41462 |
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