Theorem cnbl0 12456

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		abscl 10663	. . . . . . . . 9 |- ( x e. CC -> ( abs ` x ) e. RR )
2		absge0 10672	. . . . . . . . 9 |- ( x e. CC -> 0 <_ ( abs ` x ) )
3	1, 2	jca 302	. . . . . . . 8 |- ( x e. CC -> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) )
4	3	adantl 273	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) )
5	4	biantrurd 301	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) < R <-> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) < R ) ) )
6		df-3an 932	. . . . . 6 |- ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) <-> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) < R ) )
7	5, 6	syl6rbbr 198	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) <-> ( abs ` x ) < R ) )
8		0re 7638	. . . . . 6 |- 0 e. RR
9		simpl 108	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
10		elico2 9561	. . . . . 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 408	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,) R ) <-> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) ) )
12		0cn 7630	. . . . . . . . 9 |- 0 e. CC
13		cnblcld.1	. . . . . . . . . . 11 |- D = ( abs o. - )
14	13	cnmetdval 12451	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
15		abssub 10713	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
16	14, 15	eqtrd 2132	. . . . . . . . 9 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
17	12, 16	mpan 418	. . . . . . . 8 |- ( x e. CC -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
18		subid1 7853	. . . . . . . . 9 |- ( x e. CC -> ( x - 0 ) = x )
19	18	fveq2d 5357	. . . . . . . 8 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
20	17, 19	eqtrd 2132	. . . . . . 7 |- ( x e. CC -> ( 0 D x ) = ( abs ` x ) )
21	20	adantl 273	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( 0 D x ) = ( abs ` x ) )
22	21	breq1d 3885	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( 0 D x ) < R <-> ( abs ` x ) < R ) )
23	7, 11, 22	3bitr4d 219	. . . 4 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,) R ) <-> ( 0 D x ) < R ) )
24	23	pm5.32da 443	. . 3 |- ( R e. RR* -> ( ( x e. CC /\ ( abs ` x ) e. ( 0 [,) R ) ) <-> ( x e. CC /\ ( 0 D x ) < R ) ) )
25		absf 10722	. . . . 5 |- abs : CC --> RR
26		ffn 5208	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
27	25, 26	ax-mp 7	. . . 4 |- abs Fn CC
28		elpreima 5471	. . . 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 12453	. . . . 5 |- ( abs o. - ) e. ( *Met ` CC )
31	13, 30	eqeltri 2172	. . . 4 |- D e. ( *Met ` CC )
32		elbl 12319	. . . 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 1273	. . 3 |- ( R e. RR* -> ( x e. ( 0 ( ball ` D ) R ) <-> ( x e. CC /\ ( 0 D x ) < R ) ) )
34	24, 29, 33	3bitr4d 219	. 2 |- ( R e. RR* -> ( x e. ( `' abs " ( 0 [,) R ) ) <-> x e. ( 0 ( ball ` D ) R ) ) )
35	34	eqrdv 2098	1 |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` D ) R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 930   = wceq 1299   e. wcel 1448   class class class wbr 3875   `'ccnv 4476   "cima 4480   o. ccom 4481   Fn wfn 5054   -->wf 5055   ` cfv 5059  (class class class)co 5706   CCcc 7498   RRcr 7499   0cc0 7500   RR*cxr 7671   < clt 7672   <_ cle 7673   - cmin 7804   [,)cico 9514   abscabs 10609   *Metcxmet 11931   ballcbl 11933

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-xadd 9401  df-ico 9518  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-psmet 11938  df-xmet 11939  df-met 11940  df-bl 11941

This theorem is referenced by: (None)