ssblex - Intuitionistic Logic Explorer

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Theorem ssblex 12600

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 520	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑅 ∈ ℝ⁺)
2	1	rphalfcld 9496	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) ∈ ℝ⁺)
3		simprr 521	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ⁺)
4		rpmincl 11009	. . 3 ⊢ (((𝑅 / 2) ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ⁺)
5	2, 3, 4	syl2anc 408	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ⁺)
6	5	rpred 9483	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ)
7	2	rpred 9483	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) ∈ ℝ)
8	1	rpred 9483	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑅 ∈ ℝ)
9	3	rpred 9483	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ)
10		min1inf 11003	. . . 4 ⊢ (((𝑅 / 2) ∈ ℝ ∧ 𝑆 ∈ ℝ) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ (𝑅 / 2))
11	7, 9, 10	syl2anc 408	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ (𝑅 / 2))
12	1	rpgt0d 9486	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 0 < 𝑅)
13		halfpos 8951	. . . . 5 ⊢ (𝑅 ∈ ℝ → (0 < 𝑅 ↔ (𝑅 / 2) < 𝑅))
14	8, 13	syl 14	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (0 < 𝑅 ↔ (𝑅 / 2) < 𝑅))
15	12, 14	mpbid 146	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) < 𝑅)
16	6, 7, 8, 11, 15	lelttrd 7887	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅)
17		simpl 108	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋))
18	5	rpxrd 9484	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ^*)
19	3	rpxrd 9484	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ^*)
20		min2inf 11004	. . . 4 ⊢ (((𝑅 / 2) ∈ ℝ ∧ 𝑆 ∈ ℝ) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ 𝑆)
21	7, 9, 20	syl2anc 408	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ 𝑆)
22		ssbl 12595	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ 𝑆) → (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))
23	17, 18, 19, 21, 22	syl121anc 1221	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))
24		breq1 3932	. . . 4 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑥 < 𝑅 ↔ inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅))
25		oveq2 5782	. . . . 5 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )))
26	25	sseq1d 3126	. . . 4 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → ((𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆) ↔ (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆)))
27	24, 26	anbi12d 464	. . 3 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → ((𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)) ↔ (inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅 ∧ (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))))
28	27	rspcev 2789	. 2 ⊢ ((inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ⁺ ∧ (inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅 ∧ (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))) → ∃𝑥 ∈ ℝ⁺ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)))
29	5, 16, 23, 28	syl12anc 1214	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 set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 {cpr 3528 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 infcinf 6870 ℝcr 7619 0cc0 7620 ℝ^*cxr 7799 < clt 7800 ≤ cle 7801 / cdiv 8432 2c2 8771 ℝ⁺crp 9441 ∞Metcxmet 12149 ballcbl 12151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-xneg 9559 df-xadd 9560 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-psmet 12156 df-xmet 12157 df-bl 12159

This theorem is referenced by: mopni3 12653

W3C validator