blsscls2 - Metamath Proof Explorer

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Theorem blsscls2 24518

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 1217	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → 𝑅 < 𝑇)
3		xmetcl 24342	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) ∈ ℝ^*)
4	3	ad4ant124 1173	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → (𝑃𝐷𝑧) ∈ ℝ^*)
5		simplr1 1215	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → 𝑅 ∈ ℝ^*)
6		simplr2 1216	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) ∧ 𝑧 ∈ 𝑋) → 𝑇 ∈ ℝ^*)
7		xrlelttr 13199	. . . . . . . 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 1372	. . . . . 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 24401	. . . . . . . 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 3145	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑇 ∈ ℝ^* ∧ 𝑅 < 𝑇)) → ∀𝑧 ∈ 𝑋 ((𝑃𝐷𝑧) ≤ 𝑅 → 𝑧 ∈ (𝑃(ball‘𝐷)𝑇)))
18		rabss 4071	. . 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 4021	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 1539 ∈ wcel 2107 ∀wral 3060 {crab 3435 ⊆ wss 3950 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℝ^*cxr 11295 < clt 11296 ≤ cle 11297 ∞Metcxmet 21350 ballcbl 21352 MetOpencmopn 21355

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-psmet 21357 df-xmet 21358 df-bl 21360

This theorem is referenced by: blcld 24519 blsscls 24521 ubthlem1 30890

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