xblcntr - Metamath Proof Explorer

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Theorem xblcntr 22624

Description: A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

**Assertion**
Ref	Expression
xblcntr	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ^* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))

**Proof of Theorem xblcntr**
Step	Hyp	Ref	Expression
1		simp2 1128	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ^* ∧ 0 < 𝑅)) → 𝑃 ∈ 𝑋)
2		xmet0 22555	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃𝐷𝑃) = 0)
3	2	3adant3 1123	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ^* ∧ 0 < 𝑅)) → (𝑃𝐷𝑃) = 0)
4		simp3r 1216	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ^* ∧ 0 < 𝑅)) → 0 < 𝑅)
5	3, 4	eqbrtrd 4908	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ^* ∧ 0 < 𝑅)) → (𝑃𝐷𝑃) < 𝑅)
6		elbl 22601	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑃 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃 ∈ 𝑋 ∧ (𝑃𝐷𝑃) < 𝑅)))
7	6	3adant3r 1188	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ^* ∧ 0 < 𝑅)) → (𝑃 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃 ∈ 𝑋 ∧ (𝑃𝐷𝑃) < 𝑅)))
8	1, 5, 7	mpbir2and 703	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ^* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 0cc0 10272 ℝ^*cxr 10410 < clt 10411 ∞Metcxmet 20127 ballcbl 20129

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 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329

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 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-map 8142 df-xr 10415 df-psmet 20134 df-xmet 20135 df-bl 20137

This theorem is referenced by: blcntr 22626 xbln0 22627 blcld 22718 metds0 23061 metdseq0 23065 heicant 34070 qndenserrnbl 41439

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