Theorem cnblcld 14074

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 11121	. . . . 5 |- abs : CC --> RR
2		ffn 5367	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
3		elpreima 5637	. . . . 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 980	. . . . . . 7 |- ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) /\ ( abs ` x ) <_ R ) <-> ( ( ( abs ` x ) e. RR* /\ 0 <_ ( abs ` x ) ) /\ ( abs ` x ) <_ R ) )
6		abscl 11062	. . . . . . . . . . 11 |- ( x e. CC -> ( abs ` x ) e. RR )
7	6	rexrd 8009	. . . . . . . . . 10 |- ( x e. CC -> ( abs ` x ) e. RR* )
8		absge0 11071	. . . . . . . . . 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 8006	. . . . . . 7 |- 0 e. RR*
14		simpl 109	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
15		elicc1 9926	. . . . . . 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 7951	. . . . . . . . . 10 |- 0 e. CC
18		cnblcld.1	. . . . . . . . . . . 12 |- D = ( abs o. - )
19	18	cnmetdval 14068	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
20		abssub 11112	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( abs ` ( 0 - x ) ) = ( abs ` ( x - 0 ) ) )
21	19, 20	eqtrd 2210	. . . . . . . . . 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 8179	. . . . . . . . . 10 |- ( x e. CC -> ( x - 0 ) = x )
24	23	fveq2d 5521	. . . . . . . . 9 |- ( x e. CC -> ( abs ` ( x - 0 ) ) = ( abs ` x ) )
25	22, 24	eqtrd 2210	. . . . . . . 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 4015	. . . . . 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 2296	. 2 |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x | ( x e. CC /\ ( 0 D x ) <_ R ) } )
32		df-rab 2464	. 2 |- { x e. CC | ( 0 D x ) <_ R } = { x | ( x e. CC /\ ( 0 D x ) <_ R ) }
33	31, 32	eqtr4di 2228	1 |- ( R e. RR* -> ( `' abs " ( 0 [,] R ) ) = { x e. CC | ( 0 D x ) <_ R } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   {cab 2163   {crab 2459   class class class wbr 4005   `'ccnv 4627   "cima 4631   o. ccom 4632   Fn wfn 5213   -->wf 5214   ` cfv 5218  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   RR*cxr 7993   <_ cle 7995   - cmin 8130   [,]cicc 9893   abscabs 11008

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-rp 9656  df-icc 9897  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010

This theorem is referenced by: (None)