![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 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 24214 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))) | |
8 | 4, 5, 6, 7 | syl3anc 1368 | . . . . . . . 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 231 | . . . . . 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 1368 | . . . 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 3246 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴) |
17 | dfss3 3962 | . 2 ⊢ ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴) | |
18 | 16, 17 | sylibr 233 | 1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3053 ⊆ wss 3940 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 ℝ*cxr 11243 < clt 11244 PsMetcpsmet 21211 ballcbl 21214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-map 8817 df-xr 11248 df-psmet 21219 df-bl 21222 |
This theorem is referenced by: ioorrnopnlem 45471 |
Copyright terms: Public domain | W3C validator |