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Theorem blgt0 23834

Description: A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
blgt0	⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)

**Proof of Theorem blgt0**
Step	Hyp	Ref	Expression
1		0xr 11243	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ∈ ℝ^)
3		simpl1 1191	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝐷 ∈ (∞Met‘𝑋))
4		simpl2 1192	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑃 ∈ 𝑋)
5		elbl 23823	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
6	5	simprbda 499	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝐴 ∈ 𝑋)
7		xmetcl 23766	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑃𝐷𝐴) ∈ ℝ^*)
8	3, 4, 6, 7	syl3anc 1371	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝐴) ∈ ℝ^)
9		simpl3 1193	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 𝑅 ∈ ℝ^)
10		xmetge0 23779	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑃𝐷𝐴))
11	3, 4, 6, 10	syl3anc 1371	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 ≤ (𝑃𝐷𝐴))
12	5	simplbda 500	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → (𝑃𝐷𝐴) < 𝑅)
13	2, 8, 9, 11, 12	xrlelttrd 13121	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) ∧ 𝐴 ∈ (𝑃(ball‘𝐷)𝑅)) → 0 < 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 0cc0 11092 ℝ^*cxr 11229 < clt 11230 ≤ cle 11231 ∞Metcxmet 20863 ballcbl 20865

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-2 12257 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-psmet 20870 df-xmet 20871 df-bl 20873

This theorem is referenced by: (None)

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