ssblex - Intuitionistic Logic Explorer

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Theorem ssblex 14953

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 9844	. . 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 11599	. . 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 9831	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ)
7	2	rpred 9831	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑅 / 2) ∈ ℝ)
8	1	rpred 9831	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑅 ∈ ℝ)
9	3	rpred 9831	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ)
10		min1inf 11593	. . . 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 9834	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 0 < 𝑅)
13		halfpos 9281	. . . . 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 8210	. 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 9832	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → inf({(𝑅 / 2), 𝑆}, ℝ, < ) ∈ ℝ^*)
19	3	rpxrd 9832	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → 𝑆 ∈ ℝ^*)
20		min2inf 11594	. . . 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 14948	. . 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 1255	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ⁺ ∧ 𝑆 ∈ ℝ⁺)) → (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )) ⊆ (𝑃(ball‘𝐷)𝑆))
24		breq1 4051	. . . 4 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑥 < 𝑅 ↔ inf({(𝑅 / 2), 𝑆}, ℝ, < ) < 𝑅))
25		oveq2 5962	. . . . 5 ⊢ (𝑥 = inf({(𝑅 / 2), 𝑆}, ℝ, < ) → (𝑃(ball‘𝐷)𝑥) = (𝑃(ball‘𝐷)inf({(𝑅 / 2), 𝑆}, ℝ, < )))
26	25	sseq1d 3224	. . . 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 2879	. 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 1248	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 1373 ∈ wcel 2177 ∃wrex 2486 ⊆ wss 3168 {cpr 3636 class class class wbr 4048 ‘cfv 5277 (class class class)co 5954 infcinf 7097 ℝcr 7937 0cc0 7938 ℝ^*cxr 8119 < clt 8120 ≤ cle 8121 / cdiv 8758 2c2 9100 ℝ⁺crp 9788 ∞Metcxmet 14348 ballcbl 14350

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4164 ax-sep 4167 ax-nul 4175 ax-pow 4223 ax-pr 4258 ax-un 4485 ax-setind 4590 ax-iinf 4641 ax-cnex 8029 ax-resscn 8030 ax-1cn 8031 ax-1re 8032 ax-icn 8033 ax-addcl 8034 ax-addrcl 8035 ax-mulcl 8036 ax-mulrcl 8037 ax-addcom 8038 ax-mulcom 8039 ax-addass 8040 ax-mulass 8041 ax-distr 8042 ax-i2m1 8043 ax-0lt1 8044 ax-1rid 8045 ax-0id 8046 ax-rnegex 8047 ax-precex 8048 ax-cnre 8049 ax-pre-ltirr 8050 ax-pre-ltwlin 8051 ax-pre-lttrn 8052 ax-pre-apti 8053 ax-pre-ltadd 8054 ax-pre-mulgt0 8055 ax-pre-mulext 8056 ax-arch 8057 ax-caucvg 8058

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3001 df-csb 3096 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-nul 3463 df-if 3574 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-int 3889 df-iun 3932 df-br 4049 df-opab 4111 df-mpt 4112 df-tr 4148 df-id 4345 df-po 4348 df-iso 4349 df-iord 4418 df-on 4420 df-ilim 4421 df-suc 4423 df-iom 4644 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-res 4692 df-ima 4693 df-iota 5238 df-fun 5279 df-fn 5280 df-f 5281 df-f1 5282 df-fo 5283 df-f1o 5284 df-fv 5285 df-isom 5286 df-riota 5909 df-ov 5957 df-oprab 5958 df-mpo 5959 df-1st 6236 df-2nd 6237 df-recs 6401 df-frec 6487 df-map 6747 df-sup 7098 df-inf 7099 df-pnf 8122 df-mnf 8123 df-xr 8124 df-ltxr 8125 df-le 8126 df-sub 8258 df-neg 8259 df-reap 8661 df-ap 8668 df-div 8759 df-inn 9050 df-2 9108 df-3 9109 df-4 9110 df-n0 9309 df-z 9386 df-uz 9662 df-rp 9789 df-xneg 9907 df-xadd 9908 df-seqfrec 10606 df-exp 10697 df-cj 11203 df-re 11204 df-im 11205 df-rsqrt 11359 df-abs 11360 df-psmet 14355 df-xmet 14356 df-bl 14358

This theorem is referenced by: mopni3 15006

W3C validator