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Theorem ssbl 24454

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 1197	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝐷 ∈ (∞Met‘𝑋))
2		simp1r 1198	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑃 ∈ 𝑋)
3		simp2l 1199	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑅 ∈ ℝ^)
4		simp2r 1200	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑆 ∈ ℝ^)
5		xmet0 24373	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑃𝐷𝑃) = 0)
6	5	3ad2ant1 1133	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) = 0)
7		0re 11292	. . 3 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2852	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → (𝑃𝐷𝑃) ∈ ℝ)
9		simp3 1138	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑅 ≤ 𝑆)
10		xsubge0 13323	. . . . 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 257	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 0 ≤ (𝑆 +_𝑒 -_𝑒𝑅))
13	6, 12	eqbrtrd 5188	. 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 24433	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ⊆ wss 3976 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 ℝ^*cxr 11323 ≤ cle 11325 -_𝑒cxne 13172 +_𝑒 cxad 13173 ∞Metcxmet 21372 ballcbl 21374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-2 12356 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-psmet 21379 df-xmet 21380 df-bl 21382

This theorem is referenced by: blss 24456 ssblex 24459 blssec 24466 metequiv2 24544 met1stc 24555 met2ndci 24556 metdstri 24892 xlebnum 25016 iscmet3lem2 25345 caubl 25361 ptrecube 37580 heiborlem8 37778

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