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Theorem bl2ioo 14870
Description: A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
Hypothesis
Ref Expression
remet.1 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
Assertion
Ref Expression
bl2ioo |- ( ( A e. RR /\ B e. RR ) -> ( A ( ball ` D ) B ) = ( ( A - B ) (,) ( A + B ) ) )
Proof of Theorem bl2ioo
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 remet.1 . . . . . . . . . 10 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
21remetdval 14867 . . . . . . . . 9 |- ( ( A e. RR /\ x e. RR ) -> ( A D x ) = ( abs ` ( A - x ) ) )
3 recn 8029 . . . . . . . . . 10 |- ( A e. RR -> A e. CC )
4 recn 8029 . . . . . . . . . 10 |- ( x e. RR -> x e. CC )
5 abssub 11283 . . . . . . . . . 10 |- ( ( A e. CC /\ x e. CC ) -> ( abs ` ( A - x ) ) = ( abs ` ( x - A ) ) )
63, 4, 5syl2an 289 . . . . . . . . 9 |- ( ( A e. RR /\ x e. RR ) -> ( abs ` ( A - x ) ) = ( abs ` ( x - A ) ) )
72, 6eqtrd 2229 . . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( A D x ) = ( abs ` ( x - A ) ) )
87breq1d 4044 . . . . . . 7 |- ( ( A e. RR /\ x e. RR ) -> ( ( A D x ) < B <-> ( abs ` ( x - A ) ) < B ) )
98adantlr 477 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) -> ( ( A D x ) < B <-> ( abs ` ( x - A ) ) < B ) )
10 absdiflt 11274 . . . . . . . 8 |- ( ( x e. RR /\ A e. RR /\ B e. RR ) -> ( ( abs ` ( x - A ) ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
11103expb 1206 . . . . . . 7 |- ( ( x e. RR /\ ( A e. RR /\ B e. RR ) ) -> ( ( abs ` ( x - A ) ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
1211ancoms 268 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) -> ( ( abs ` ( x - A ) ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
139, 12bitrd 188 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) -> ( ( A D x ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
1413pm5.32da 452 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( x e. RR /\ ( A D x ) < B ) <-> ( x e. RR /\ ( ( A - B ) < x /\ x < ( A + B ) ) ) ) )
15 3anass 984 . . . 4 |- ( ( x e. RR /\ ( A - B ) < x /\ x < ( A + B ) ) <-> ( x e. RR /\ ( ( A - B ) < x /\ x < ( A + B ) ) ) )
1614, 15bitr4di 198 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( x e. RR /\ ( A D x ) < B ) <-> ( x e. RR /\ ( A - B ) < x /\ x < ( A + B ) ) ) )
17 rexr 8089 . . . 4 |- ( B e. RR -> B e. RR* )
181rexmet 14869 . . . . 5 |- D e. ( *Met ` RR )
19 elbl 14711 . . . . 5 |- ( ( D e. ( *Met ` RR ) /\ A e. RR /\ B e. RR* ) -> ( x e. ( A ( ball ` D ) B ) <-> ( x e. RR /\ ( A D x ) < B ) ) )
2018, 19mp3an1 1335 . . . 4 |- ( ( A e. RR /\ B e. RR* ) -> ( x e. ( A ( ball ` D ) B ) <-> ( x e. RR /\ ( A D x ) < B ) ) )
2117, 20sylan2 286 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A ( ball ` D ) B ) <-> ( x e. RR /\ ( A D x ) < B ) ) )
22 resubcl 8307 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
23 readdcl 8022 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
24 rexr 8089 . . . . 5 |- ( ( A - B ) e. RR -> ( A - B ) e. RR* )
25 rexr 8089 . . . . 5 |- ( ( A + B ) e. RR -> ( A + B ) e. RR* )
26 elioo2 10013 . . . . 5 |- ( ( ( A - B ) e. RR* /\ ( A + B ) e. RR* ) -> ( x e. ( ( A - B ) (,) ( A + B ) ) <-> ( x e. RR /\ ( A - B ) < x /\ x < ( A + B ) ) ) )
2724, 25, 26syl2an 289 . . . 4 |- ( ( ( A - B ) e. RR /\ ( A + B ) e. RR ) -> ( x e. ( ( A - B ) (,) ( A + B ) ) <-> ( x e. RR /\ ( A - B ) < x /\ x < ( A + B ) ) ) )
2822, 23, 27syl2anc 411 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( ( A - B ) (,) ( A + B ) ) <-> ( x e. RR /\ ( A - B ) < x /\ x < ( A + B ) ) ) )
2916, 21, 283bitr4d 220 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A ( ball ` D ) B ) <-> x e. ( ( A - B ) (,) ( A + B ) ) ) )
3029eqrdv 2194 1 |- ( ( A e. RR /\ B e. RR ) -> ( A ( ball ` D ) B ) = ( ( A - B ) (,) ( A + B ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4034   X. cxp 4662   |` cres 4666   o. ccom 4668   ` cfv 5259  (class class class)co 5925   CCcc 7894   RRcr 7895   + caddc 7899   RR*cxr 8077   < clt 8078   - cmin 8214   (,)cioo 9980   abscabs 11179   *Metcxmet 14168   ballcbl 14170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-xadd 9865  df-ioo 9984  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178
This theorem is referenced by:  ioo2bl  14871  blssioo  14873  tgioo  14874
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