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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 | β’ (((π· β (PsMetβπ) β§ π β β+) β§ (π΄ β π β§ π΅ β π)) β (π΅ β (π΄(ballβπ·)π ) β π΅(β‘π· β (0[,)π ))π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpxr 12990 | . . 3 β’ (π β β+ β π β β*) | |
2 | blcomps 24219 | . . 3 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π΄ β π β§ π΅ β π)) β (π΅ β (π΄(ballβπ·)π ) β π΄ β (π΅(ballβπ·)π ))) | |
3 | 1, 2 | sylanl2 678 | . 2 β’ (((π· β (PsMetβπ) β§ π β β+) β§ (π΄ β π β§ π΅ β π)) β (π΅ β (π΄(ballβπ·)π ) β π΄ β (π΅(ballβπ·)π ))) |
4 | simpll 764 | . . 3 β’ (((π· β (PsMetβπ) β§ π β β+) β§ (π΄ β π β§ π΅ β π)) β π· β (PsMetβπ)) | |
5 | simprr 770 | . . 3 β’ (((π· β (PsMetβπ) β§ π β β+) β§ (π΄ β π β§ π΅ β π)) β π΅ β π) | |
6 | simplr 766 | . . 3 β’ (((π· β (PsMetβπ) β§ π β β+) β§ (π΄ β π β§ π΅ β π)) β π β β+) | |
7 | blval2 24391 | . . . 4 β’ ((π· β (PsMetβπ) β§ π΅ β π β§ π β β+) β (π΅(ballβπ·)π ) = ((β‘π· β (0[,)π )) β {π΅})) | |
8 | 7 | eleq2d 2818 | . . 3 β’ ((π· β (PsMetβπ) β§ π΅ β π β§ π β β+) β (π΄ β (π΅(ballβπ·)π ) β π΄ β ((β‘π· β (0[,)π )) β {π΅}))) |
9 | 4, 5, 6, 8 | syl3anc 1370 | . 2 β’ (((π· β (PsMetβπ) β§ π β β+) β§ (π΄ β π β§ π΅ β π)) β (π΄ β (π΅(ballβπ·)π ) β π΄ β ((β‘π· β (0[,)π )) β {π΅}))) |
10 | elimasng 6087 | . . . . 5 β’ ((π΅ β π β§ π΄ β π) β (π΄ β ((β‘π· β (0[,)π )) β {π΅}) β β¨π΅, π΄β© β (β‘π· β (0[,)π )))) | |
11 | df-br 5149 | . . . . 5 β’ (π΅(β‘π· β (0[,)π ))π΄ β β¨π΅, π΄β© β (β‘π· β (0[,)π ))) | |
12 | 10, 11 | bitr4di 289 | . . . 4 β’ ((π΅ β π β§ π΄ β π) β (π΄ β ((β‘π· β (0[,)π )) β {π΅}) β π΅(β‘π· β (0[,)π ))π΄)) |
13 | 12 | ancoms 458 | . . 3 β’ ((π΄ β π β§ π΅ β π) β (π΄ β ((β‘π· β (0[,)π )) β {π΅}) β π΅(β‘π· β (0[,)π ))π΄)) |
14 | 13 | adantl 481 | . 2 β’ (((π· β (PsMetβπ) β§ π β β+) β§ (π΄ β π β§ π΅ β π)) β (π΄ β ((β‘π· β (0[,)π )) β {π΅}) β π΅(β‘π· β (0[,)π ))π΄)) |
15 | 3, 9, 14 | 3bitrd 305 | 1 β’ (((π· β (PsMetβπ) β§ π β β+) β§ (π΄ β π β§ π΅ β π)) β (π΅ β (π΄(ballβπ·)π ) β π΅(β‘π· β (0[,)π ))π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 β wcel 2105 {csn 4628 β¨cop 4634 class class class wbr 5148 β‘ccnv 5675 β cima 5679 βcfv 6543 (class class class)co 7412 0cc0 11116 β*cxr 11254 β+crp 12981 [,)cico 13333 PsMetcpsmet 21217 ballcbl 21220 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 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-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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-2 12282 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-psmet 21225 df-bl 21228 |
This theorem is referenced by: metucn 24400 |
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