blsscls2 - Metamath Proof Explorer

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Theorem blsscls2 24392

Description: A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)

**Hypotheses**
Ref	Expression
mopni.1	⊢ 𝐽 = (MetOpen‘𝐷)
blcld.3	⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}

**Assertion**
Ref	Expression
blsscls2	⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) → 𝑆 ⊆ (𝑃(ball‘𝐷)𝑇))

Distinct variable groups: 𝑧,𝐷 𝑧,𝑅 𝑧,𝑃 𝑧,𝑇 𝑧,𝑋 Allowed substitution hints: 𝑆(𝑧) 𝐽(𝑧)

**Proof of Theorem blsscls2**
Step	Hyp	Ref	Expression
1		blcld.3	. 2 ⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}
2		simplr3 1218	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → 𝑅 < 𝑇)
3		xmetcl 24219	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) ∈ ℝ^*)
4	3	ad4ant124 1174	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) ∈ ℝ^*)
5		simplr1 1216	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → 𝑅 ∈ ℝ^*)
6		simplr2 1217	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → 𝑇 ∈ ℝ^*)
7		xrlelttr 13116	. . . . . . . 8 ⊢ (((𝑃𝐷𝑧) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^*) → (((𝑃𝐷𝑧) ≤ 𝑅 ∧ 𝑅 < 𝑇) → (𝑃𝐷𝑧) < 𝑇))
8	7	expcomd 416	. . . . . . 7 ⊢ (((𝑃𝐷𝑧) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^*) → (𝑅 < 𝑇 → ((𝑃𝐷𝑧) ≤ 𝑅 → (𝑃𝐷𝑧) < 𝑇)))
9	4, 5, 6, 8	syl3anc 1373	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → (𝑅 < 𝑇 → ((𝑃𝐷𝑧) ≤ 𝑅 → (𝑃𝐷𝑧) < 𝑇)))
10	2, 9	mpd 15	. . . . 5 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐷𝑧) ≤ 𝑅 → (𝑃𝐷𝑧) < 𝑇))
11		simp2 1137	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇) → 𝑇 ∈ ℝ^*)
12		elbl2 24278	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑇 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧 ∈ (𝑃(ball‘𝐷)𝑇) ↔ (𝑃𝐷𝑧) < 𝑇))
13	12	an4s 660	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑇 ∈ ℝ^* ∧ 𝑧 ∈ 𝑋)) → (𝑧 ∈ (𝑃(ball‘𝐷)𝑇) ↔ (𝑃𝐷𝑧) < 𝑇))
14	11, 13	sylanr1 682	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ ((𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇) ∧ 𝑧 ∈ 𝑋)) → (𝑧 ∈ (𝑃(ball‘𝐷)𝑇) ↔ (𝑃𝐷𝑧) < 𝑇))
15	14	anassrs 467	. . . . 5 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ (𝑃(ball‘𝐷)𝑇) ↔ (𝑃𝐷𝑧) < 𝑇))
16	10, 15	sylibrd 259	. . . 4 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝑃(ball‘𝐷)𝑇)))
17	16	ralrimiva 3125	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) → ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝑃(ball‘𝐷)𝑇)))
18		rabss 4035	. . 3 ⊢ ({𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝑃(ball‘𝐷)𝑇) ↔ ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝑃(ball‘𝐷)𝑇)))
19	17, 18	sylibr 234	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) → {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅} ⊆ (𝑃(ball‘𝐷)𝑇))
20	1, 19	eqsstrid 3985	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) → 𝑆 ⊆ (𝑃(ball‘𝐷)𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 {crab 3405 ⊆ wss 3914 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 ∞Metcxmet 21249 ballcbl 21251 MetOpencmopn 21254

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-pre-lttri 11142 ax-pre-lttrn 11143

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-psmet 21256 df-xmet 21257 df-bl 21259

This theorem is referenced by: blcld 24393 blsscls 24395 ubthlem1 30799

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