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 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝜑) | |
3 | simpr 477 | . . . . . . 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 22192 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))) | |
8 | 4, 5, 6, 7 | syl3anc 1326 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))) |
9 | 8 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅))) |
10 | 3, 9 | mpbid 222 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)) |
11 | 10 | simpld 475 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝑋) |
12 | 10 | simprd 479 | . . . . 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 1326 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑥 ∈ 𝐴) |
15 | 14 | ex 450 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) → 𝑥 ∈ 𝐴)) |
16 | 1, 15 | ralrimi 2957 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴) |
17 | dfss3 3592 | . 2 ⊢ ((𝑃(ball‘𝐷)𝑅) ⊆ 𝐴 ↔ ∀𝑥 ∈ (𝑃(ball‘𝐷)𝑅)𝑥 ∈ 𝐴) | |
18 | 16, 17 | sylibr 224 | 1 ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝ*cxr 10073 < clt 10074 PsMetcpsmet 19730 ballcbl 19733 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-xr 10078 df-psmet 19738 df-bl 19741 |
This theorem is referenced by: ioorrnopnlem 40524 |
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