ssblex - Intuitionistic Logic Explorer

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Theorem ssblex 15148

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 9937	. . 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 11792	. . 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 9924	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ)
7	2	rpred 9924	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) ∈ ℝ)
8	1	rpred 9924	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑅 ∈ ℝ)
9	3	rpred 9924	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ)
10		min1inf 11786	. . . 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 9927	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 0 < 𝑅)
13		halfpos 9368	. . . . 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 8297	. 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 9925	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ^*)
19	3	rpxrd 9925	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ^*)
20		min2inf 11787	. . . 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 15143	. . 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 1276	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))
24		breq1 4089	. . . 4 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑥 < 𝑅 ↔ inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅))
25		oveq2 6021	. . . . 5 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )))
26	25	sseq1d 3254	. . . 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 2908	. 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 1269	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 1395 ∈ wcel 2200 ∃wrex 2509 ⊆ wss 3198 {cpr 3668 class class class wbr 4086 ‘cfv 5324 (class class class)co 6013 infcinf 7176 ℝcr 8024 0cc0 8025 ℝ^*cxr 8206 < clt 8207 ≤ cle 8208 / cdiv 8845 2c2 9187 ℝ⁺crp 9881 ∞Metcxmet 14543 ballcbl 14545

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-map 6814 df-sup 7177 df-inf 7178 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-n0 9396 df-z 9473 df-uz 9749 df-rp 9882 df-xneg 10000 df-xadd 10001 df-seqfrec 10703 df-exp 10794 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-psmet 14550 df-xmet 14551 df-bl 14553

This theorem is referenced by: mopni3 15201

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