Theorem cnbl0 14702

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 11195	. . . . . . . . 9 |- ( x e. CC -> ( abs ` x ) e. RR )
3		absge0 11204	. . . . . . . . 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 8019	. . . . . 6 |- 0 e. RR
9		simpl 109	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
10		elico2 10003	. . . . . 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 8011	. . . . . . . . 9 |- 0 e. CC
13		cnblcld.1	. . . . . . . . . . 11 |- D = ( abs o. - )
14	13	cnmetdval 14697	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
15		abssub 11245	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
16	14, 15	eqtrd 2226	. . . . . . . . 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 8239	. . . . . . . . 9 |- ( x e. CC -> ( x - 0 ) = x )
19	18	fveq2d 5558	. . . . . . . 8 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
20	17, 19	eqtrd 2226	. . . . . . 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 4039	. . . . 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 11254	. . . . 5 |- abs : CC --> RR
26		ffn 5403	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
27	25, 26	ax-mp 5	. . . 4 |- abs Fn CC
28		elpreima 5677	. . . 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 14699	. . . . 5 |- ( abs o. - ) e. ( *Met ` CC )
31	13, 30	eqeltri 2266	. . . 4 |- D e. ( *Met ` CC )
32		elbl 14559	. . . 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 2191	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 2164   class class class wbr 4029   `'ccnv 4658   "cima 4662   o. ccom 4663   Fn wfn 5249   -->wf 5250   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   RR*cxr 8053   < clt 8054   <_ cle 8055   - cmin 8190   [,)cico 9956   abscabs 11141   *Metcxmet 14032   ballcbl 14034

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-xadd 9839  df-ico 9960  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-psmet 14039  df-xmet 14040  df-met 14041  df-bl 14042

This theorem is referenced by: (None)