Theorem cnmet 14850

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 8020	. 2 |- CC e. _V
2		absf 11292	. . 3 |- abs : CC --> RR
3		subf 8245	. . 3 |- - : ( CC X. CC ) --> CC
4		fco 5426	. . 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 8242	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x - y ) e. CC )
7	6	abs00ad 11247	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( ( abs ` ( x - y ) ) = 0 <-> ( x - y ) = 0 ) )
8		eqid 2196	. . . . . 6 |- ( abs o. - ) = ( abs o. - )
9	8	cnmetdval 14849	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x ( abs o. - ) y ) = ( abs ` ( x - y ) ) )
10	9	eqcomd 2202	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( abs ` ( x - y ) ) = ( x ( abs o. - ) y ) )
11	10	eqeq1d 2205	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( ( abs ` ( x - y ) ) = 0 <-> ( x ( abs o. - ) y ) = 0 ) )
12		subeq0 8269	. . 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 11287	. . . 4 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( abs ` ( x - y ) ) <_ ( ( abs ` ( x - z ) ) + ( abs ` ( z - y ) ) ) )
15		abssub 11283	. . . . . 6 |- ( ( x e. CC /\ z e. CC ) -> ( abs ` ( x - z ) ) = ( abs ` ( z - x ) ) )
16	15	oveq1d 5940	. . . . 5 |- ( ( x e. CC /\ z e. CC ) -> ( ( abs ` ( x - z ) ) + ( abs ` ( z - y ) ) ) = ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
17	16	3adant2 1018	. . . 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 4060	. . 3 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( abs ` ( x - y ) ) <_ ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
19	9	3adant3 1019	. . 3 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x ( abs o. - ) y ) = ( abs ` ( x - y ) ) )
20	8	cnmetdval 14849	. . . . . 6 |- ( ( z e. CC /\ x e. CC ) -> ( z ( abs o. - ) x ) = ( abs ` ( z - x ) ) )
21	20	3adant3 1019	. . . . 5 |- ( ( z e. CC /\ x e. CC /\ y e. CC ) -> ( z ( abs o. - ) x ) = ( abs ` ( z - x ) ) )
22	8	cnmetdval 14849	. . . . . 6 |- ( ( z e. CC /\ y e. CC ) -> ( z ( abs o. - ) y ) = ( abs ` ( z - y ) ) )
23	22	3adant2 1018	. . . . 5 |- ( ( z e. CC /\ x e. CC /\ y e. CC ) -> ( z ( abs o. - ) y ) = ( abs ` ( z - y ) ) )
24	21, 23	oveq12d 5943	. . . 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 1212	. . 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 4066	. 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 14666	1 |- ( abs o. - ) e. ( Met ` CC )

Syntax hints:   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   X. cxp 4662   o. ccom 4668   -->wf 5255   ` cfv 5259  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   + caddc 7899   <_ cle 8079   - cmin 8214   abscabs 11179   Metcmet 14169

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

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 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-met 14177

This theorem is referenced by:  cnxmet  14851  cnfldms  14856  remet  14868