Theorem cnbl0 14854

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 982	. . . . . 6 |- ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) <-> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) < R ) )
2		abscl 11233	. . . . . . . . 9 |- ( x e. CC -> ( abs ` x ) e. RR )
3		absge0 11242	. . . . . . . . 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 8043	. . . . . 6 |- 0 e. RR
9		simpl 109	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
10		elico2 10029	. . . . . 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 414	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,) R ) <-> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) ) )
12		0cn 8035	. . . . . . . . 9 |- 0 e. CC
13		cnblcld.1	. . . . . . . . . . 11 |- D = ( abs o. - )
14	13	cnmetdval 14849	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
15		abssub 11283	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
16	14, 15	eqtrd 2229	. . . . . . . . 9 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
17	12, 16	mpan 424	. . . . . . . 8 |- ( x e. CC -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
18		subid1 8263	. . . . . . . . 9 |- ( x e. CC -> ( x - 0 ) = x )
19	18	fveq2d 5565	. . . . . . . 8 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
20	17, 19	eqtrd 2229	. . . . . . 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 4044	. . . . 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 452	. . 3 |- ( R e. RR* -> ( ( x e. CC /\ ( abs ` x ) e. ( 0 [,) R ) ) <-> ( x e. CC /\ ( 0 D x ) < R ) ) )
25		absf 11292	. . . . 5 |- abs : CC --> RR
26		ffn 5410	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
27	25, 26	ax-mp 5	. . . 4 |- abs Fn CC
28		elpreima 5684	. . . 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 14851	. . . . 5 |- ( abs o. - ) e. ( *Met ` CC )
31	13, 30	eqeltri 2269	. . . 4 |- D e. ( *Met ` CC )
32		elbl 14711	. . . 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 1338	. . 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 2194	1 |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` D ) R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4034   `'ccnv 4663   "cima 4667   o. ccom 4668   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   RR*cxr 8077   < clt 8078   <_ cle 8079   - cmin 8214   [,)cico 9982   abscabs 11179   *Metcxmet 14168   ballcbl 14170

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-xadd 9865  df-ico 9986  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178

This theorem is referenced by: (None)