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Theorem bl2ioo 15293
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 15290 . . . . . . . . 9 |- ( ( A e. RR /\ x e. RR ) -> ( A D x ) = ( abs ` ( A - x ) ) )
3 recn 8165 . . . . . . . . . 10 |- ( A e. RR -> A e. CC )
4 recn 8165 . . . . . . . . . 10 |- ( x e. RR -> x e. CC )
5 abssub 11679 . . . . . . . . . 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 2264 . . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( A D x ) = ( abs ` ( x - A ) ) )
87breq1d 4098 . . . . . . 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 11670 . . . . . . . 8 |- ( ( x e. RR /\ A e. RR /\ B e. RR ) -> ( ( abs ` ( x - A ) ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
11103expb 1230 . . . . . . 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 1008 . . . 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 8225 . . . 4 |- ( B e. RR -> B e. RR* )
181rexmet 15292 . . . . 5 |- D e. ( *Met ` RR )
19 elbl 15134 . . . . 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 1360 . . . 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 8443 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
23 readdcl 8158 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
24 rexr 8225 . . . . 5 |- ( ( A - B ) e. RR -> ( A - B ) e. RR* )
25 rexr 8225 . . . . 5 |- ( ( A + B ) e. RR -> ( A + B ) e. RR* )
26 elioo2 10156 . . . . 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 2229 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 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   X. cxp 4723   |` cres 4727   o. ccom 4729   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   + caddc 8035   RR*cxr 8213   < clt 8214   - cmin 8350   (,)cioo 10123   abscabs 11575   *Metcxmet 14569   ballcbl 14571
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-rp 9889  df-xadd 10008  df-ioo 10127  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-psmet 14576  df-xmet 14577  df-met 14578  df-bl 14579
This theorem is referenced by:  ioo2bl  15294  blssioo  15296  tgioo  15297
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