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Theorem ssblex 23934

Description: A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

**Assertion**
Ref	Expression
ssblex	⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → ∃𝑥 ∈ ℝ⁺ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)))

Distinct variable groups: 𝑥,𝐷 𝑥,𝑅 𝑥,𝑃 𝑥,𝑆 𝑥,𝑋

**Proof of Theorem ssblex**
Step	Hyp	Ref	Expression
1		simprl 770	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑅 ∈ ℝ⁺)
2	1	rphalfcld 13028	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) ∈ ℝ⁺)
3		simprr 772	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ⁺)
4	2, 3	ifcld 4575	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ∈ ℝ⁺)
5	4	rpred 13016	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ∈ ℝ)
6	2	rpred 13016	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) ∈ ℝ)
7	1	rpred 13016	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑅 ∈ ℝ)
8	3	rpred 13016	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ)
9		min1 13168	. . . 4 ⊢ (((𝑅 / 2) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ≤ (𝑅 / 2))
10	6, 8, 9	syl2anc 585	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ≤ (𝑅 / 2))
11	1	rpgt0d 13019	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 0 < 𝑅)
12		halfpos 12442	. . . . 5 ⊢ (𝑅 ∈ ℝ → (0 < 𝑅 ↔ (𝑅 / 2) < 𝑅))
13	7, 12	syl 17	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (0 < 𝑅 ↔ (𝑅 / 2) < 𝑅))
14	11, 13	mpbid 231	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) < 𝑅)
15	5, 6, 7, 10, 14	lelttrd 11372	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) < 𝑅)
16		simpl 484	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋))
17	4	rpxrd 13017	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ∈ ℝ^*)
18	3	rpxrd 13017	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ^*)
19		min2 13169	. . . 4 ⊢ (((𝑅 / 2) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ≤ 𝑆)
20	6, 8, 19	syl2anc 585	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ≤ 𝑆)
21		ssbl 23929	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ≤ 𝑆) → (𝑃(ball‘𝐷)if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆)) ⊆ (𝑃(ball‘𝐷)𝑆))
22	16, 17, 18, 20, 21	syl121anc 1376	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑃(ball‘𝐷)if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆)) ⊆ (𝑃(ball‘𝐷)𝑆))
23		breq1 5152	. . . 4 ⊢ (𝑥 = if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) → (𝑥 < 𝑅 ↔ if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) < 𝑅))
24		oveq2 7417	. . . . 5 ⊢ (𝑥 = if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆)))
25	24	sseq1d 4014	. . . 4 ⊢ (𝑥 = if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆) ↔ (𝑃(ball‘𝐷)if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆)) ⊆ (𝑃(ball‘𝐷)𝑆)))
26	23, 25	anbi12d 632	. . 3 ⊢ (𝑥 = if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) → ((𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)) ↔ (if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) < 𝑅 ∧ (𝑃(ball‘𝐷)if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆)) ⊆ (𝑃(ball‘𝐷)𝑆))))
27	26	rspcev 3613	. 2 ⊢ ((if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) ∈ ℝ⁺ ∧ (if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆) < 𝑅 ∧ (𝑃(ball‘𝐷)if((𝑅 / 2) ≤ 𝑆, (𝑅 / 2), 𝑆)) ⊆ (𝑃(ball‘𝐷)𝑆))) → ∃𝑥 ∈ ℝ⁺ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)))
28	4, 15, 22, 27	syl12anc 836	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → ∃𝑥 ∈ ℝ⁺ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 ⊆ wss 3949 ifcif 4529 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 / cdiv 11871 2c2 12267 ℝ⁺crp 12974 ∞Metcxmet 20929 ballcbl 20931

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-2 12275 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-psmet 20936 df-xmet 20937 df-bl 20939

This theorem is referenced by: mopni3 24003

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