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Theorem bl2ioo 15264
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 15261 . . . . . . . . 9 |- ( ( A e. RR /\ x e. RR ) -> ( A D x ) = ( abs ` ( A - x ) ) )
3 recn 8155 . . . . . . . . . 10 |- ( A e. RR -> A e. CC )
4 recn 8155 . . . . . . . . . 10 |- ( x e. RR -> x e. CC )
5 abssub 11652 . . . . . . . . . 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 2262 . . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( A D x ) = ( abs ` ( x - A ) ) )
87breq1d 4096 . . . . . . 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 11643 . . . . . . . 8 |- ( ( x e. RR /\ A e. RR /\ B e. RR ) -> ( ( abs ` ( x - A ) ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
11103expb 1228 . . . . . . 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 1006 . . . 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 8215 . . . 4 |- ( B e. RR -> B e. RR* )
181rexmet 15263 . . . . 5 |- D e. ( *Met ` RR )
19 elbl 15105 . . . . 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 1358 . . . 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 8433 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
23 readdcl 8148 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
24 rexr 8215 . . . . 5 |- ( ( A - B ) e. RR -> ( A - B ) e. RR* )
25 rexr 8215 . . . . 5 |- ( ( A + B ) e. RR -> ( A + B ) e. RR* )
26 elioo2 10146 . . . . 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 2227 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 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086   X. cxp 4721   |` cres 4725   o. ccom 4727   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   + caddc 8025   RR*cxr 8203   < clt 8204   - cmin 8340   (,)cioo 10113   abscabs 11548   *Metcxmet 14540   ballcbl 14542
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-map 6814  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-rp 9879  df-xadd 9998  df-ioo 10117  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-psmet 14547  df-xmet 14548  df-met 14549  df-bl 14550
This theorem is referenced by:  ioo2bl  15265  blssioo  15267  tgioo  15268
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