Theorem cnblcld 14771

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 11275	. . . . 5 |- abs : CC --> RR
2		ffn 5407	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
3		elpreima 5681	. . . . 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 982	. . . . . . 7 |- ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) <-> ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) <_ R ) )
6		abscl 11216	. . . . . . . . . . 11 |- ( x e. CC -> ( abs ` x ) e. RR )
7	6	rexrd 8076	. . . . . . . . . 10 |- ( x e. CC -> ( abs ` x ) e. RR* )
8		absge0 11225	. . . . . . . . . 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 8073	. . . . . . 7 |- 0 e. RR*
14		simpl 109	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
15		elicc1 9999	. . . . . . 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 8018	. . . . . . . . . 10 |- 0 e. CC
18		cnblcld.1	. . . . . . . . . . . 12 |- D = ( abs o. - )
19	18	cnmetdval 14765	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
20		abssub 11266	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
21	19, 20	eqtrd 2229	. . . . . . . . . 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 8246	. . . . . . . . . 10 |- ( x e. CC -> ( x - 0 ) = x )
24	23	fveq2d 5562	. . . . . . . . 9 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
25	22, 24	eqtrd 2229	. . . . . . . 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 4043	. . . . . 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 2315	. 2 |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x | ( x e. CC /\ ( 0 D x ) <_ R ) } )
32		df-rab 2484	. 2 |- { x e. CC | ( 0 D x ) <_ R } = { x | ( x e. CC /\ ( 0 D x ) <_ R ) }
33	31, 32	eqtr4di 2247	1 |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x e. CC | ( 0 D x ) <_ R } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   {cab 2182   {crab 2479   class class class wbr 4033   `'ccnv 4662   "cima 4666   o. ccom 4667   Fn wfn 5253   -->wf 5254   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   RR*cxr 8060   <_ cle 8062   - cmin 8197   [,]cicc 9966   abscabs 11162

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-icc 9970  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164

This theorem is referenced by: (None)