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Theorem ssbl 24358

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 1198	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝐷 ∈ (∞Met‘𝑋))
2		simp1r 1199	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^*) ∧ 𝑅 ≤ 𝑆) → 𝑃 ∈ 𝑋)
3		simp2l 1200	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑅 ∈ ℝ^)
4		simp2r 1201	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^) ∧ 𝑅 ≤ 𝑆) → 𝑆 ∈ ℝ^)
5		xmet0 24277	. . . 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 11125	. . 3 ⊢ 0 ∈ ℝ
8	6, 7	eqeltrdi 2841	. 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 13167	. . . . 5 ⊢ ((𝑆 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (0 ≤ (𝑆 +_𝑒 -_𝑒𝑅) ↔ 𝑅 ≤ 𝑆))
11	4, 3, 10	syl2anc 584	. . . 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 5117	. 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 24337	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 1086 = wceq 1541 ∈ wcel 2113 ⊆ wss 3898 class class class wbr 5095 ‘cfv 6489 (class class class)co 7355 ℝcr 11016 0cc0 11017 ℝ^*cxr 11156 ≤ cle 11158 -_𝑒cxne 13014 +_𝑒 cxad 13015 ∞Metcxmet 21285 ballcbl 21287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-1st 7930 df-2nd 7931 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-2 12199 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-psmet 21292 df-xmet 21293 df-bl 21295

This theorem is referenced by: blss 24360 ssblex 24363 blssec 24370 metequiv2 24445 met1stc 24456 met2ndci 24457 metdstri 24787 xlebnum 24911 iscmet3lem2 25239 caubl 25255 ptrecube 37733 heiborlem8 37931

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