Theorem cnmet 15117

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 8084	. 2 |- CC e. _V
2		absf 11536	. . 3 |- abs : CC --> RR
3		subf 8309	. . 3 |- - : ( CC X. CC ) --> CC
4		fco 5461	. . 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 8306	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x - y ) e. CC )
7	6	abs00ad 11491	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( ( abs ` ( x - y ) ) = 0 <-> ( x - y ) = 0 ) )
8		eqid 2207	. . . . . 6 |- ( abs o. - ) = ( abs o. - )
9	8	cnmetdval 15116	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x ( abs o. - ) y ) = ( abs ` ( x - y ) ) )
10	9	eqcomd 2213	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( abs ` ( x - y ) ) = ( x ( abs o. - ) y ) )
11	10	eqeq1d 2216	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( ( abs ` ( x - y ) ) = 0 <-> ( x ( abs o. - ) y ) = 0 ) )
12		subeq0 8333	. . 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 11531	. . . 4 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( abs ` ( x - y ) ) <_ ( ( abs ` ( x - z ) ) + ( abs ` ( z - y ) ) ) )
15		abssub 11527	. . . . . 6 |- ( ( x e. CC /\ z e. CC ) -> ( abs ` ( x - z ) ) = ( abs ` ( z - x ) ) )
16	15	oveq1d 5982	. . . . 5 |- ( ( x e. CC /\ z e. CC ) -> ( ( abs ` ( x - z ) ) + ( abs ` ( z - y ) ) ) = ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
17	16	3adant2 1019	. . . 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 4085	. . 3 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( abs ` ( x - y ) ) <_ ( ( abs ` ( z - x ) ) + ( abs ` ( z - y ) ) ) )
19	9	3adant3 1020	. . 3 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x ( abs o. - ) y ) = ( abs ` ( x - y ) ) )
20	8	cnmetdval 15116	. . . . . 6 |- ( ( z e. CC /\ x e. CC ) -> ( z ( abs o. - ) x ) = ( abs ` ( z - x ) ) )
21	20	3adant3 1020	. . . . 5 |- ( ( z e. CC /\ x e. CC /\ y e. CC ) -> ( z ( abs o. - ) x ) = ( abs ` ( z - x ) ) )
22	8	cnmetdval 15116	. . . . . 6 |- ( ( z e. CC /\ y e. CC ) -> ( z ( abs o. - ) y ) = ( abs ` ( z - y ) ) )
23	22	3adant2 1019	. . . . 5 |- ( ( z e. CC /\ x e. CC /\ y e. CC ) -> ( z ( abs o. - ) y ) = ( abs ` ( z - y ) ) )
24	21, 23	oveq12d 5985	. . . 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 1213	. . 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 4091	. 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 14933	1 |- ( abs o. - ) e. ( Met ` CC )

Syntax hints:   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   X. cxp 4691   o. ccom 4697   -->wf 5286   ` cfv 5290  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   + caddc 7963   <_ cle 8143   - cmin 8278   abscabs 11423   Metcmet 14414

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-map 6760  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-met 14422

This theorem is referenced by:  cnxmet  15118  cnfldms  15123  remet  15135