elbl2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > elbl2		Structured version Visualization version GIF version

Theorem elbl2 23002

Description: Membership in a ball. (Contributed by NM, 9-Mar-2007.)

**Assertion**
Ref	Expression
elbl2	⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))

**Proof of Theorem elbl2**
Step	Hyp	Ref	Expression
1		simprr 771	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
2		elbl 23000	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
3	2	3expa 1114	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑅 ∈ ℝ^*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
4	3	an32s 650	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ 𝑃 ∈ 𝑋) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
5	4	adantrr 715	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)))
6	1, 5	mpbirand 705	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ^*) ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑃𝐷𝐴) < 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝ^*cxr 10676 < clt 10677 ∞Metcxmet 20532 ballcbl 20534

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-xr 10681 df-psmet 20539 df-xmet 20540 df-bl 20542

This theorem is referenced by: elbl3 23004 blcom 23006 imasf1obl 23100 prdsbl 23103 blsscls2 23116 metcnp 23153 ngpocelbl 23315 zdis 23426 metdsge 23459 cfil3i 23874 iscfil3 23878 iscmet3lem2 23897 caubl 23913 dvlog2lem 25237 lgamucov 25617 isbnd3 35064 cntotbnd 35076 ismtyima 35083 stirlinglem5 42370 hoiqssbllem2 42912

W3C validator