ssblex - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ssblex		GIF version

Theorem ssblex 13598

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 529	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑅 ∈ ℝ⁺)
2	1	rphalfcld 9696	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) ∈ ℝ⁺)
3		simprr 531	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ⁺)
4		rpmincl 11230	. . 3 ⊢ (((𝑅 / 2) ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ⁺)
5	2, 3, 4	syl2anc 411	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ⁺)
6	5	rpred 9683	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ)
7	2	rpred 9683	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) ∈ ℝ)
8	1	rpred 9683	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑅 ∈ ℝ)
9	3	rpred 9683	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ)
10		min1inf 11224	. . . 4 ⊢ (((𝑅 / 2) ∈ ℝ ∧ 𝑆 ∈ ℝ) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ (𝑅 / 2))
11	7, 9, 10	syl2anc 411	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ (𝑅 / 2))
12	1	rpgt0d 9686	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 0 < 𝑅)
13		halfpos 9139	. . . . 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 147	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) < 𝑅)
16	6, 7, 8, 11, 15	lelttrd 8072	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅)
17		simpl 109	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋))
18	5	rpxrd 9684	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ^*)
19	3	rpxrd 9684	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ^*)
20		min2inf 11225	. . . 4 ⊢ (((𝑅 / 2) ∈ ℝ ∧ 𝑆 ∈ ℝ) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ 𝑆)
21	7, 9, 20	syl2anc 411	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ≤ 𝑆)
22		ssbl 13593	. . 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 1243	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))
24		breq1 4003	. . . 4 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑥 < 𝑅 ↔ inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅))
25		oveq2 5877	. . . . 5 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )))
26	25	sseq1d 3184	. . . 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 473	. . 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 2841	. 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 1236	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 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ⊆ wss 3129 {cpr 3592 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 infcinf 6976 ℝcr 7801 0cc0 7802 ℝ^*cxr 7981 < clt 7982 ≤ cle 7983 / cdiv 8618 2c2 8959 ℝ⁺crp 9640 ∞Metcxmet 13147 ballcbl 13149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-map 6644 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-rp 9641 df-xneg 9759 df-xadd 9760 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-psmet 13154 df-xmet 13155 df-bl 13157

This theorem is referenced by: mopni3 13651

W3C validator