Theorem cnbl0 13075

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 969	. . . . . 6 |- ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) <-> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) < R ) )
2		abscl 10979	. . . . . . . . 9 |- ( x e. CC -> ( abs ` x ) e. RR )
3		absge0 10988	. . . . . . . . 9 |- ( x e. CC -> 0 <_ ( abs ` x ) )
4	2, 3	jca 304	. . . . . . . 8 |- ( x e. CC -> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) )
5	4	adantl 275	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) )
6	5	biantrurd 303	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) < R <-> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) < R ) ) )
7	1, 6	bitr4id 198	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) <-> ( abs ` x ) < R ) )
8		0re 7890	. . . . . 6 |- 0 e. RR
9		simpl 108	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
10		elico2 9864	. . . . . 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 411	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,) R ) <-> ( ( abs ` x ) e. RR /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) < R ) ) )
12		0cn 7882	. . . . . . . . 9 |- 0 e. CC
13		cnblcld.1	. . . . . . . . . . 11 |- D = ( abs o. - )
14	13	cnmetdval 13070	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
15		abssub 11029	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
16	14, 15	eqtrd 2197	. . . . . . . . 9 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
17	12, 16	mpan 421	. . . . . . . 8 |- ( x e. CC -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
18		subid1 8109	. . . . . . . . 9 |- ( x e. CC -> ( x - 0 ) = x )
19	18	fveq2d 5484	. . . . . . . 8 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
20	17, 19	eqtrd 2197	. . . . . . 7 |- ( x e. CC -> ( 0 D x ) = ( abs ` x ) )
21	20	adantl 275	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( 0 D x ) = ( abs ` x ) )
22	21	breq1d 3986	. . . . 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 448	. . 3 |- ( R e. RR* -> ( ( x e. CC /\ ( abs ` x ) e. ( 0 [,) R ) ) <-> ( x e. CC /\ ( 0 D x ) < R ) ) )
25		absf 11038	. . . . 5 |- abs : CC --> RR
26		ffn 5331	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
27	25, 26	ax-mp 5	. . . 4 |- abs Fn CC
28		elpreima 5598	. . . 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 13072	. . . . 5 |- ( abs o. - ) e. ( *Met ` CC )
31	13, 30	eqeltri 2237	. . . 4 |- D e. ( *Met ` CC )
32		elbl 12932	. . . 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 1316	. . 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 2162	1 |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` D ) R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 967   = wceq 1342   e. wcel 2135   class class class wbr 3976   `'ccnv 4597   "cima 4601   o. ccom 4602   Fn wfn 5177   -->wf 5178   ` cfv 5182  (class class class)co 5836   CCcc 7742   RRcr 7743   0cc0 7744   RR*cxr 7923   < clt 7924   <_ cle 7925   - cmin 8060   [,)cico 9817   abscabs 10925   *Metcxmet 12521   ballcbl 12523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862  ax-arch 7863  ax-caucvg 7864

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-map 6607  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-rp 9581  df-xadd 9700  df-ico 9821  df-seqfrec 10371  df-exp 10445  df-cj 10770  df-re 10771  df-im 10772  df-rsqrt 10926  df-abs 10927  df-psmet 12528  df-xmet 12529  df-met 12530  df-bl 12531

This theorem is referenced by: (None)