Theorem cnblcld 15249

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 11661	. . . . 5 |- abs : CC --> RR
2		ffn 5479	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
3		elpreima 5762	. . . . 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		df-3an 1004	. . . . . . 7 |- ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) <-> ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) <_ R ) )
6		abscl 11602	. . . . . . . . . . 11 |- ( x e. CC -> ( abs ` x ) e. RR )
7	6	rexrd 8219	. . . . . . . . . 10 |- ( x e. CC -> ( abs ` x ) e. RR* )
8		absge0 11611	. . . . . . . . . 10 |- ( x e. CC -> 0 <_ ( abs ` x ) )
9	7, 8	jca 306	. . . . . . . . 9 |- ( x e. CC -> ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) )
10	9	adantl 277	. . . . . . . 8 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) )
11	10	biantrurd 305	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) <_ R <-> ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) <_ R ) ) )
12	5, 11	bitr4id 199	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) <-> ( abs ` x ) <_ R ) )
13		0xr 8216	. . . . . . 7 |- 0 e. RR*
14		simpl 109	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
15		elicc1 10149	. . . . . . 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 414	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,] R ) <-> ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) ) )
17		0cn 8161	. . . . . . . . . 10 |- 0 e. CC
18		cnblcld.1	. . . . . . . . . . . 12 |- D = ( abs o. - )
19	18	cnmetdval 15243	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
20		abssub 11652	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
21	19, 20	eqtrd 2262	. . . . . . . . . 10 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
22	17, 21	mpan 424	. . . . . . . . 9 |- ( x e. CC -> ( 0 D x ) = ( abs ` ( x - 0 ) ) )
23		subid1 8389	. . . . . . . . . 10 |- ( x e. CC -> ( x - 0 ) = x )
24	23	fveq2d 5639	. . . . . . . . 9 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
25	22, 24	eqtrd 2262	. . . . . . . 8 |- ( x e. CC -> ( 0 D x ) = ( abs ` x ) )
26	25	adantl 277	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> ( 0 D x ) = ( abs ` x ) )
27	26	breq1d 4096	. . . . . 6 |- ( ( R e. RR* /\ x e. CC ) -> ( ( 0 D x ) <_ R <-> ( abs ` x ) <_ R ) )
28	12, 16, 27	3bitr4d 220	. . . . 5 |- ( ( R e. RR* /\ x e. CC ) -> ( ( abs ` x ) e. ( 0 [,] R ) <-> ( 0 D x ) <_ R ) )
29	28	pm5.32da 452	. . . 4 |- ( R e. RR* -> ( ( x e. CC /\ ( abs ` x ) e. ( 0 [,] R ) ) <-> ( x e. CC /\ ( 0 D x ) <_ R ) ) )
30	4, 29	bitrid 192	. . 3 |- ( R e. RR* -> ( x e. ( `' abs " ( 0 [,] R ) ) <-> ( x e. CC /\ ( 0 D x ) <_ R ) ) )
31	30	abbi2dv 2348	. 2 |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x | ( x e. CC /\ ( 0 D x ) <_ R ) } )
32		df-rab 2517	. 2 |- { x e. CC | ( 0 D x ) <_ R } = { x | ( x e. CC /\ ( 0 D x ) <_ R ) }
33	31, 32	eqtr4di 2280	1 |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x e. CC | ( 0 D x ) <_ R } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   {crab 2512   class class class wbr 4086   `'ccnv 4722   "cima 4726   o. ccom 4727   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   RR*cxr 8203   <_ cle 8205   - cmin 8340   [,]cicc 10116   abscabs 11548

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-rp 9879  df-icc 10120  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550

This theorem is referenced by: (None)