ssblps - Intuitionistic Logic Explorer

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Theorem ssblps 12597

Description: The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)

**Assertion**
Ref	Expression
ssblps	⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))

**Proof of Theorem ssblps**
Step	Hyp	Ref	Expression
1		simp1l 1005	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝐷 ∈ (PsMet‘𝑋))
2		simp1r 1006	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑃 ∈ 𝑋)
3		simp2l 1007	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑅 ∈ ℝ^)
4		simp2r 1008	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑆 ∈ ℝ^)
5		psmet0 12499	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃𝐷𝑃) = 0)
6	5	3ad2ant1 1002	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) = 0)
7		0re 7769	. . 3 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2230	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) ∈ ℝ)
9		simp3 983	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑅 ≤ 𝑆)
10		xsubge0 9667	. . . . 5 ⊢ ((𝑆 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 ≤ (𝑆 +_𝑒 -_𝑒𝑅) ↔ 𝑅 ≤ 𝑆))
11	4, 3, 10	syl2anc 408	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (0 ≤ (𝑆 +_𝑒 -_𝑒𝑅) ↔ 𝑅 ≤ 𝑆))
12	9, 11	mpbird 166	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 0 ≤ (𝑆 +_𝑒 -_𝑒𝑅))
13	6, 12	eqbrtrd 3950	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) ≤ (𝑆 +_𝑒 -_𝑒𝑅))
14	1, 2, 2, 3, 4, 8, 13	xblss2ps 12576	1 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ⊆ wss 3071 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℝcr 7622 0cc0 7623 ℝ^*cxr 7802 ≤ cle 7804 -_𝑒cxne 9559 +_𝑒 cxad 9560 PsMetcpsmet 12151 ballcbl 12154

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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 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-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 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-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-2 8782 df-xneg 9562 df-xadd 9563 df-psmet 12159 df-bl 12162

This theorem is referenced by: blssps 12599

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