Theorem cnblcld 13702

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 11103	. . . . 5 |- abs : CC --> RR
2		ffn 5361	. . . . 5 |- ( abs : CC --> RR -> abs Fn CC )
3		elpreima 5631	. . . . 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 11044	. . . . . . . . . . 11 |- ( x e. CC -> ( abs ` x ) e. RR )
7	6	rexrd 7997	. . . . . . . . . 10 |- ( x e. CC -> ( abs ` x ) e. RR* )
8		absge0 11053	. . . . . . . . . 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 7994	. . . . . . 7 |- 0 e. RR*
14		simpl 109	. . . . . . 7 |- ( ( R e. RR* /\ x e. CC ) -> R e. RR* )
15		elicc1 9911	. . . . . . 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 7940	. . . . . . . . . 10 |- 0 e. CC
18		cnblcld.1	. . . . . . . . . . . 12 |- D = ( abs o. - )
19	18	cnmetdval 13696	. . . . . . . . . . 11 |- ( ( 0 e. CC /\ x e. CC ) -> ( 0 D x ) = ( abs ` ( 0 - x ) ) )
20		abssub 11094	. . . . . . . . . . 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 8167	. . . . . . . . . 10 |- ( x e. CC -> ( x - 0 ) = x )
24	23	fveq2d 5515	. . . . . . . . 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 4010	. . . . . 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 4000   `'ccnv 4622   "cima 4626   o. ccom 4627   Fn wfn 5207   -->wf 5208   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   RR*cxr 7981   <_ cle 7983   - cmin 8118   [,]cicc 9878   abscabs 10990

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-rp 9641  df-icc 9882  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992

This theorem is referenced by: (None)