ssblps - Metamath Proof Explorer

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Theorem ssblps 23025

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 1192	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝐷 ∈ (PsMet‘𝑋))
2		simp1r 1193	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑃 ∈ 𝑋)
3		simp2l 1194	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑅 ∈ ℝ^)
4		simp2r 1195	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑆 ∈ ℝ^)
5		psmet0 22911	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃𝐷𝑃) = 0)
6	5	3ad2ant1 1128	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) = 0)
7		0re 10636	. . 3 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2920	. 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) ∈ ℝ)
9		simp3 1133	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑅 ≤ 𝑆)
10		xsubge0 12648	. . . . 5 ⊢ ((𝑆 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 ≤ (𝑆 +_𝑒 -_𝑒𝑅) ↔ 𝑅 ≤ 𝑆))
11	4, 3, 10	syl2anc 586	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (0 ≤ (𝑆 +_𝑒 -_𝑒𝑅) ↔ 𝑅 ≤ 𝑆))
12	9, 11	mpbird 259	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 0 ≤ (𝑆 +_𝑒 -_𝑒𝑅))
13	6, 12	eqbrtrd 5081	. 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 23004	1 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 ℝcr 10529 0cc0 10530 ℝ^*cxr 10667 ≤ cle 10669 -_𝑒cxne 12498 +_𝑒 cxad 12499 PsMetcpsmet 20522 ballcbl 20525

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-2 11694 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-psmet 20530 df-bl 20533

This theorem is referenced by: blssps 23027

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