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Mirrors > Home > MPE Home > Th. List > elbl4 | Structured version Visualization version GIF version |
Description: Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
Ref | Expression |
---|---|
elbl4 | ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ (𝐴(ball‘𝐷)𝑅) ↔ 𝐵(◡𝐷 “ (0[,)𝑅))𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpxr 12619 | . . 3 ⊢ (𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ*) | |
2 | blcomps 23315 | . . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ (𝐴(ball‘𝐷)𝑅) ↔ 𝐴 ∈ (𝐵(ball‘𝐷)𝑅))) | |
3 | 1, 2 | sylanl2 681 | . 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ (𝐴(ball‘𝐷)𝑅) ↔ 𝐴 ∈ (𝐵(ball‘𝐷)𝑅))) |
4 | simpll 767 | . . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐷 ∈ (PsMet‘𝑋)) | |
5 | simprr 773 | . . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋) | |
6 | simplr 769 | . . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑅 ∈ ℝ+) | |
7 | blval2 23484 | . . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝐵(ball‘𝐷)𝑅) = ((◡𝐷 “ (0[,)𝑅)) “ {𝐵})) | |
8 | 7 | eleq2d 2824 | . . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝐴 ∈ (𝐵(ball‘𝐷)𝑅) ↔ 𝐴 ∈ ((◡𝐷 “ (0[,)𝑅)) “ {𝐵}))) |
9 | 4, 5, 6, 8 | syl3anc 1373 | . 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ∈ (𝐵(ball‘𝐷)𝑅) ↔ 𝐴 ∈ ((◡𝐷 “ (0[,)𝑅)) “ {𝐵}))) |
10 | elimasng 5970 | . . . . 5 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ ((◡𝐷 “ (0[,)𝑅)) “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ (◡𝐷 “ (0[,)𝑅)))) | |
11 | df-br 5068 | . . . . 5 ⊢ (𝐵(◡𝐷 “ (0[,)𝑅))𝐴 ↔ 〈𝐵, 𝐴〉 ∈ (◡𝐷 “ (0[,)𝑅))) | |
12 | 10, 11 | bitr4di 292 | . . . 4 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ ((◡𝐷 “ (0[,)𝑅)) “ {𝐵}) ↔ 𝐵(◡𝐷 “ (0[,)𝑅))𝐴)) |
13 | 12 | ancoms 462 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∈ ((◡𝐷 “ (0[,)𝑅)) “ {𝐵}) ↔ 𝐵(◡𝐷 “ (0[,)𝑅))𝐴)) |
14 | 13 | adantl 485 | . 2 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ∈ ((◡𝐷 “ (0[,)𝑅)) “ {𝐵}) ↔ 𝐵(◡𝐷 “ (0[,)𝑅))𝐴)) |
15 | 3, 9, 14 | 3bitrd 308 | 1 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ (𝐴(ball‘𝐷)𝑅) ↔ 𝐵(◡𝐷 “ (0[,)𝑅))𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2111 {csn 4555 〈cop 4561 class class class wbr 5067 ◡ccnv 5564 “ cima 5568 ‘cfv 6397 (class class class)co 7231 0cc0 10753 ℝ*cxr 10890 ℝ+crp 12610 [,)cico 12961 PsMetcpsmet 20371 ballcbl 20374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-po 5482 df-so 5483 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-1st 7779 df-2nd 7780 df-er 8411 df-map 8530 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-div 11514 df-2 11917 df-rp 12611 df-xneg 12728 df-xadd 12729 df-xmul 12730 df-ico 12965 df-psmet 20379 df-bl 20382 |
This theorem is referenced by: metucn 23493 |
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