Theorem cnbl0 15635

Description: Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)

Hypothesis

Ref	Expression
cnblcld.1	|- D = ( abs o. - )

Assertion

Ref	Expression
cnbl0	|- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` D ) R ) )

Proof of Theorem cnbl0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 1011	. . . . . 6 |- ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) <-> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) < R ) )
2		abscl 11817	. . . . . . . . 9 |- ( x e. CC -> ( abs ` x ) e. RR )
3		absge0 11826	. . . . . . . . 9 |- ( x e. CC -> 0 <_ ( abs ` x ) )
4	2, 3	jca 306	. . . . . . . 8 |- ( x e. CC -> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) )
5	4	adantl 277	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) )
6	5	biantrurd 305	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) < R <-> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) < R ) ) )
7	1, 6	bitr4id 199	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) <-> ( abs ` x ) < R ) )
8		0re 8326	. . . . . 6 |- 0 e. RR
9		simpl 109	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
10		elico2 10339	. . . . . 6 |- ( ( 0 e. RR /\ R e. RR* ) -> ( ( abs ` x ) e. ( 0 [,) R ) <-> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) ) )
11	8, 9, 10	sylancr 418	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,) R ) <-> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) ) )
12		0cn 8318	. . . . . . . . 9 |- 0 e. CC
13		cnblcld.1	. . . . . . . . . . 11 |- D = ( abs o. - )
14	13	cnmetdval 15630	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
15		abssub 11867	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
16	14, 15	eqtrd 2271	. . . . . . . . 9 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
17	12, 16	mpan 428	. . . . . . . 8 |- ( x e. CC -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
18		subid1 8546	. . . . . . . . 9 |- ( x e. CC -> ( x - 0 ) = x )
19	18	fveq2d 5699	. . . . . . . 8 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
20	17, 19	eqtrd 2271	. . . . . . 7 |- ( x e. CC -> ( 0 D x ) = ( abs ` x ) )
21	20	adantl 277	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( 0 D x ) = ( abs ` x ) )
22	21	breq1d 4140	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( 0 D x ) < R <-> ( abs ` x ) < R ) )
23	7, 11, 22	3bitr4d 220	. . . 4 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,) R ) <-> ( 0 D x ) < R ) )
24	23	pm5.32da 456	. . 3 |- ( R e. RR* -> ( ( x e. CC /\ ( abs ` x ) e. ( 0 [,) R ) ) <-> ( x e. CC /\ ( 0 D x ) < R ) ) )
25		absf 11876	. . . . 5 |- abs : CC --> RR
26		ffn 5533	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
27	25, 26	ax-mp 5	. . . 4 |- abs Fn CC
28		elpreima 5828	. . . 4 |- ( abs Fn CC -> ( x e. ( `' abs " ( 0 [,) R ) ) <-> ( x e. CC /\ ( abs ` x ) e. ( 0 [,) R ) ) ) )
29	27, 28	mp1i 10	. . 3 |- ( R e. RR* -> ( x e. ( `' abs " ( 0 [,) R ) ) <-> ( x e. CC /\ ( abs ` x ) e. ( 0 [,) R ) ) ) )
30		cnxmet 15632	. . . . 5 |- ( abs o. - ) e. ( *Met ` CC )
31	13, 30	eqeltri 2311	. . . 4 |- D e. ( *Met ` CC )
32		elbl 15492	. . . 4 |- ( ( D e. ( *Met ` CC ) /\ 0 e. CC /\ R e. RR* ) -> ( x e. ( 0 ( ball ` D ) R ) <-> ( x e. CC /\ ( 0 D x ) < R ) ) )
33	31, 12, 32	mp3an12 1368	. . 3 |- ( R e. RR* -> ( x e. ( 0 ( ball ` D ) R ) <-> ( x e. CC /\ ( 0 D x ) < R ) ) )
34	24, 29, 33	3bitr4d 220	. 2 |- ( R e. RR* -> ( x e. ( `' abs " ( 0 [,) R ) ) <-> x e. ( 0 ( ball ` D ) R ) ) )
35	34	eqrdv 2236	1 |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` D ) R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   `'ccnv 4773   "cima 4777   o. ccom 4778   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   RR*cxr 8359   < clt 8360   <_ cle 8361   - cmin 8497   [,)cico 10292   abscabs 11763   *Metcxmet 14873   ballcbl 14875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-xadd 10175  df-ico 10296  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883

This theorem is used by: (None)