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Theorem bl2ioo 15097
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 15094 . . . . . . . . 9 |- ( ( A e. RR /\ x e. RR ) -> ( A D x ) = ( abs ` ( A - x ) ) )
3 recn 8078 . . . . . . . . . 10 |- ( A e. RR -> A e. CC )
4 recn 8078 . . . . . . . . . 10 |- ( x e. RR -> x e. CC )
5 abssub 11487 . . . . . . . . . 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 2239 . . . . . . . 8 |- ( ( A e. RR /\ x e. RR ) -> ( A D x ) = ( abs ` ( x - A ) ) )
87breq1d 4061 . . . . . . 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 11478 . . . . . . . 8 |- ( ( x e. RR /\ A e. RR /\ B e. RR ) -> ( ( abs ` ( x - A ) ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
11103expb 1207 . . . . . . 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 985 . . . 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 8138 . . . 4 |- ( B e. RR -> B e. RR* )
181rexmet 15096 . . . . 5 |- D e. ( *Met ` RR )
19 elbl 14938 . . . . 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 1337 . . . 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 8356 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
23 readdcl 8071 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
24 rexr 8138 . . . . 5 |- ( ( A - B ) e. RR -> ( A - B ) e. RR* )
25 rexr 8138 . . . . 5 |- ( ( A + B ) e. RR -> ( A + B ) e. RR* )
26 elioo2 10063 . . . . 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 2204 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 981   = wceq 1373   e. wcel 2177   class class class wbr 4051   X. cxp 4681   |` cres 4685   o. ccom 4687   ` cfv 5280  (class class class)co 5957   CCcc 7943   RRcr 7944   + caddc 7948   RR*cxr 8126   < clt 8127   - cmin 8263   (,)cioo 10030   abscabs 11383   *Metcxmet 14373   ballcbl 14375
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-map 6750  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-rp 9796  df-xadd 9915  df-ioo 10034  df-seqfrec 10615  df-exp 10706  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-psmet 14380  df-xmet 14381  df-met 14382  df-bl 14383
This theorem is referenced by:  ioo2bl  15098  blssioo  15100  tgioo  15101
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