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Theorem xrsblre 24547

Description: Any ball of the metric of the extended reals centered on an element of ℝ is entirely contained in ℝ. (Contributed by Mario Carneiro, 4-Sep-2015.)

**Hypothesis**
Ref	Expression
xrsxmet.1	⊢ 𝐷 = (dist‘ℝ^*_𝑠)

**Assertion**
Ref	Expression
xrsblre	⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝐷)𝑅) ⊆ ℝ)

Proof of Theorem xrsblre Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexr 11264	. . 3 ⊢ (𝑃 ∈ ℝ → 𝑃 ∈ ℝ^*)
2		xrsxmet.1	. . . . 5 ⊢ 𝐷 = (dist‘ℝ^*_𝑠)
3	2	xrsxmet 24545	. . . 4 ⊢ 𝐷 ∈ (∞Met‘ℝ^*)
4		eqid 2732	. . . . 5 ⊢ (^◡𝐷 “ ℝ) = (^◡𝐷 “ ℝ)
5	4	blssec 24161	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑃 ∈ ℝ^ ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](^◡𝐷 “ ℝ))
6	3, 5	mp3an1 1448	. . 3 ⊢ ((𝑃 ∈ ℝ^* ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](^◡𝐷 “ ℝ))
7	1, 6	sylan 580	. 2 ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](^◡𝐷 “ ℝ))
8		vex 3478	. . . . 5 ⊢ 𝑥 ∈ V
9		simpl 483	. . . . 5 ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → 𝑃 ∈ ℝ)
10		elecg 8748	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑃 ∈ ℝ) → (𝑥 ∈ [𝑃](^◡𝐷 “ ℝ) ↔ 𝑃(^◡𝐷 “ ℝ)𝑥))
11	8, 9, 10	sylancr 587	. . . 4 ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ [𝑃](^◡𝐷 “ ℝ) ↔ 𝑃(^◡𝐷 “ ℝ)𝑥))
12	4	xmeterval 24158	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘ℝ^) → (𝑃(^◡𝐷 “ ℝ)𝑥 ↔ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)))
13	3, 12	ax-mp 5	. . . . 5 ⊢ (𝑃(^◡𝐷 “ ℝ)𝑥 ↔ (𝑃 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ))
14		simpr 485	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 = 𝑥) → 𝑃 = 𝑥)
15		simplll 773	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 = 𝑥) → 𝑃 ∈ ℝ)
16	14, 15	eqeltrrd 2834	. . . . . . 7 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 = 𝑥) → 𝑥 ∈ ℝ)
17		simplr3 1217	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 ≠ 𝑥) → (𝑃𝐷𝑥) ∈ ℝ)
18		simplr1 1215	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 ≠ 𝑥) → 𝑃 ∈ ℝ^*)
19		simplr2 1216	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 ≠ 𝑥) → 𝑥 ∈ ℝ^*)
20		simpr 485	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 ≠ 𝑥) → 𝑃 ≠ 𝑥)
21	2	xrsdsreclb 21192	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑃 ≠ 𝑥) → ((𝑃𝐷𝑥) ∈ ℝ ↔ (𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ)))
22	18, 19, 20, 21	syl3anc 1371	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 ≠ 𝑥) → ((𝑃𝐷𝑥) ∈ ℝ ↔ (𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ)))
23	17, 22	mpbid 231	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 ≠ 𝑥) → (𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ))
24	23	simprd 496	. . . . . . 7 ⊢ ((((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) ∧ 𝑃 ≠ 𝑥) → 𝑥 ∈ ℝ)
25	16, 24	pm2.61dane 3029	. . . . . 6 ⊢ (((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) ∧ (𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ)) → 𝑥 ∈ ℝ)
26	25	ex 413	. . . . 5 ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^) → ((𝑃 ∈ ℝ^ ∧ 𝑥 ∈ ℝ^* ∧ (𝑃𝐷𝑥) ∈ ℝ) → 𝑥 ∈ ℝ))
27	13, 26	biimtrid 241	. . . 4 ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → (𝑃(^◡𝐷 “ ℝ)𝑥 → 𝑥 ∈ ℝ))
28	11, 27	sylbid 239	. . 3 ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ [𝑃](^◡𝐷 “ ℝ) → 𝑥 ∈ ℝ))
29	28	ssrdv 3988	. 2 ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → [𝑃](^◡𝐷 “ ℝ) ⊆ ℝ)
30	7, 29	sstrd 3992	1 ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝐷)𝑅) ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ⊆ wss 3948 class class class wbr 5148 ^◡ccnv 5675 “ cima 5679 ‘cfv 6543 (class class class)co 7411 [cec 8703 ℝcr 11111 ℝ^*cxr 11251 distcds 17210 ℝ^*_𝑠cxrs 17450 ∞Metcxmet 21129 ballcbl 21131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-ec 8707 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-icc 13335 df-fz 13489 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-mulr 17215 df-tset 17220 df-ple 17221 df-ds 17223 df-xrs 17452 df-psmet 21136 df-xmet 21137 df-bl 21139

This theorem is referenced by: xrsmopn 24548

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