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Theorem ssbl 23484

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

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

**Proof of Theorem ssbl**
Step	Hyp	Ref	Expression
1		simp1l 1195	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝐷 ∈ (∞Met‘𝑋))
2		simp1r 1196	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑃 ∈ 𝑋)
3		simp2l 1197	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑅 ∈ ℝ^)
4		simp2r 1198	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑆 ∈ ℝ^)
5		xmet0 23403	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃𝐷𝑃) = 0)
6	5	3ad2ant1 1131	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) = 0)
7		0re 10908	. . 3 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2847	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) ∈ ℝ)
9		simp3 1136	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑅 ≤ 𝑆)
10		xsubge0 12924	. . . . 5 ⊢ ((𝑆 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 ≤ (𝑆 +_𝑒 -_𝑒𝑅) ↔ 𝑅 ≤ 𝑆))
11	4, 3, 10	syl2anc 583	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (0 ≤ (𝑆 +_𝑒 -_𝑒𝑅) ↔ 𝑅 ≤ 𝑆))
12	9, 11	mpbird 256	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 0 ≤ (𝑆 +_𝑒 -_𝑒𝑅))
13	6, 12	eqbrtrd 5092	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) ≤ (𝑆 +_𝑒 -_𝑒𝑅))
14	1, 2, 2, 3, 4, 8, 13	xblss2 23463	1 ⊢ (((𝐷 ∈ (∞Met‘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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 ℝ^*cxr 10939 ≤ cle 10941 -_𝑒cxne 12774 +_𝑒 cxad 12775 ∞Metcxmet 20495 ballcbl 20497

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-2 11966 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-psmet 20502 df-xmet 20503 df-bl 20505

This theorem is referenced by: blss 23486 ssblex 23489 blssec 23496 metequiv2 23572 met1stc 23583 met2ndci 23584 metdstri 23920 xlebnum 24034 iscmet3lem2 24361 caubl 24377 ptrecube 35704 heiborlem8 35903

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