Theorem cnmet 15395

Description: The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)

Assertion

Ref	Expression
cnmet	|- ( abs o. - ) e. ( Met ` CC )

Proof of Theorem cnmet

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 8251	. 2 |- CC e. _V
2		absf 11795	. . 3 |- abs : CC --> RR
3		subf 8475	. . 3 |- - : ( CC X. CC ) --> CC
4		fco 5527	. . 3 |- ( ( abs : CC --> RR /\ - : ( CC X. CC ) --> CC ) -> ( abs o. - ) : ( CC X. CC ) --> RR )
5	2, 3, 4	mp2an 426	. 2 |- ( abs o. - ) : ( CC X. CC ) --> RR
6		subcl 8472	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x - y ) e. CC )
7	6	abs00ad 11750	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( ( abs ` ( x - y ) ) = 0 <-> ( x - y ) = 0 ) )
8		eqid 2232	. . . . . 6 |- ( abs o. - ) = ( abs o. - )
9	8	cnmetdval 15394	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x ( abs o. - ) y ) = ( abs ` ( x - y ) ) )
10	9	eqcomd 2238	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( abs ` ( x - y ) ) = ( x ( abs o. - ) y ) )
11	10	eqeq1d 2241	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( ( abs ` ( x - y ) ) = 0 <-> ( x ( abs o. - ) y ) = 0 ) )
12		subeq0 8499	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( ( x - y ) = 0 <-> x = y ) )
13	7, 11, 12	3bitr3d 218	. 2 |- ( ( x e. CC /\ y e. CC ) -> ( ( x ( abs o. - ) y ) = 0 <-> x = y ) )
14		abs3dif 11790	. . . 4 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( abs ` ( x - y ) ) <_ ( ( abs ` ( x - z ) ) + ( abs ` ( z - y ) ) ) )
15		abssub 11786	. . . . . 6 |- ( ( x e. CC /\ z e. CC ) -> ( abs ` ( x - z ) ) = ( abs ` ( z - x ) ) )
16	15	oveq1d 6065	. . . . 5 |- ( ( x e. CC /\ z e. CC ) -> ( ( abs ` ( x - z ) ) + ( abs ` ( z - y ) ) ) = ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
17	16	3adant2 1043	. . . 4 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( abs ` ( x - z ) ) + ( abs ` ( z - y ) ) ) = ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
18	14, 17	breqtrd 4135	. . 3 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( abs ` ( x - y ) ) <_ ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
19	9	3adant3 1044	. . 3 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x ( abs o. - ) y ) = ( abs ` ( x - y ) ) )
20	8	cnmetdval 15394	. . . . . 6 |- ( ( z e. CC /\ x e. CC ) -> ( z ( abs o. - ) x ) = ( abs ` ( z - x ) ) )
21	20	3adant3 1044	. . . . 5 |- ( ( z e. CC /\ x e. CC /\ y e. CC ) -> ( z ( abs o. - ) x ) = ( abs ` ( z - x ) ) )
22	8	cnmetdval 15394	. . . . . 6 |- ( ( z e. CC /\ y e. CC ) -> ( z ( abs o. - ) y ) = ( abs ` ( z - y ) ) )
23	22	3adant2 1043	. . . . 5 |- ( ( z e. CC /\ x e. CC /\ y e. CC ) -> ( z ( abs o. - ) y ) = ( abs ` ( z - y ) ) )
24	21, 23	oveq12d 6068	. . . 4 |- ( ( z e. CC /\ x e. CC /\ y e. CC ) -> ( ( z ( abs o. - ) x ) + ( z ( abs o. - ) y ) ) = ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
25	24	3coml 1237	. . 3 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( ( z ( abs o. - ) x ) + ( z ( abs o. - ) y ) ) = ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
26	18, 19, 25	3brtr4d 4141	. 2 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x ( abs o. - ) y ) <_ ( ( z ( abs o. - ) x ) + ( z ( abs o. - ) y ) ) )
27	1, 5, 13, 26	ismeti 15211	1 |- ( abs o. - ) e. ( Met ` CC )

Syntax hints:   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   X. cxp 4747   o. ccom 4753   -->wf 5348   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   + caddc 8130   <_ cle 8309   - cmin 8444   abscabs 11682   Metcmet 14685

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-map 6884  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-rp 9987  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-met 14693

This theorem is referenced by:  cnxmet  15396  cnfldms  15401  remet  15413