Theorem cnblcld 12704

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

Hypothesis

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

Assertion

Ref	Expression
cnblcld	|- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x e. CC | ( 0 D x ) <_ R } )

Distinct variable groups:   x, D   x, R

Proof of Theorem cnblcld

Step	Hyp	Ref	Expression
1		absf 10882	. . . . 5 |- abs : CC --> RR
2		ffn 5272	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
3		elpreima 5539	. . . . 5 |- ( abs Fn CC -> ( x e. ( `' abs " ( 0 [,] R ) ) <-> ( x e. CC /\ ( abs ` x ) e. ( 0 [,] R ) ) ) )
4	1, 2, 3	mp2b 8	. . . 4 |- ( x e. ( `' abs " ( 0 [,] R ) ) <-> ( x e. CC /\ ( abs ` x ) e. ( 0 [,] R ) ) )
5		abscl 10823	. . . . . . . . . . 11 |- ( x e. CC -> ( abs ` x ) e. RR )
6	5	rexrd 7815	. . . . . . . . . 10 |- ( x e. CC -> ( abs ` x ) e. RR* )
7		absge0 10832	. . . . . . . . . 10 |- ( x e. CC -> 0 <_ ( abs ` x ) )
8	6, 7	jca 304	. . . . . . . . 9 |- ( x e. CC -> ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) )
9	8	adantl 275	. . . . . . . 8 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) )
10	9	biantrurd 303	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) <_ R <-> ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) <_ R ) ) )
11		df-3an 964	. . . . . . 7 |- ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) <-> ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) <_ R ) )
12	10, 11	syl6rbbr 198	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) <-> ( abs ` x ) <_ R ) )
13		0xr 7812	. . . . . . 7 |- 0 e. RR*
14		simpl 108	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
15		elicc1 9707	. . . . . . 7 |- ( ( 0 e. RR* /\ R e. RR* ) -> ( ( abs ` x ) e. ( 0 [,] R ) <-> ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) ) )
16	13, 14, 15	sylancr 410	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,] R ) <-> ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) ) )
17		0cn 7758	. . . . . . . . . 10 |- 0 e. CC
18		cnblcld.1	. . . . . . . . . . . 12 |- D = ( abs o. - )
19	18	cnmetdval 12698	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
20		abssub 10873	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
21	19, 20	eqtrd 2172	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
22	17, 21	mpan 420	. . . . . . . . 9 |- ( x e. CC -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
23		subid1 7982	. . . . . . . . . 10 |- ( x e. CC -> ( x - 0 ) = x )
24	23	fveq2d 5425	. . . . . . . . 9 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
25	22, 24	eqtrd 2172	. . . . . . . 8 |- ( x e. CC -> ( 0 D x ) = ( abs ` x ) )
26	25	adantl 275	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> ( 0 D x ) = ( abs ` x ) )
27	26	breq1d 3939	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( 0 D x ) <_ R <-> ( abs ` x ) <_ R ) )
28	12, 16, 27	3bitr4d 219	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,] R ) <-> ( 0 D x ) <_ R ) )
29	28	pm5.32da 447	. . . 4 |- ( R e. RR* -> ( ( x e. CC /\ ( abs ` x ) e. ( 0 [,] R ) ) <-> ( x e. CC /\ ( 0 D x ) <_ R ) ) )
30	4, 29	syl5bb 191	. . 3 |- ( R e. RR* -> ( x e. ( `' abs " ( 0 [,] R ) ) <-> ( x e. CC /\ ( 0 D x ) <_ R ) ) )
31	30	abbi2dv 2258	. 2 |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x | ( x e. CC /\ ( 0 D x ) <_ R ) } )
32		df-rab 2425	. 2 |- { x e. CC | ( 0 D x ) <_ R } = { x | ( x e. CC /\ ( 0 D x ) <_ R ) }
33	31, 32	syl6eqr 2190	1 |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x e. CC | ( 0 D x ) <_ R } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2125   {crab 2420   class class class wbr 3929   `'ccnv 4538   "cima 4542   o. ccom 4543   Fn wfn 5118   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   RR*cxr 7799   <_ cle 7801   - cmin 7933   [,]cicc 9674   abscabs 10769

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-rp 9442  df-icc 9678  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771

This theorem is referenced by: (None)