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Theorem cnmet 20351
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.
StepHypRef Expression
1 cnex 9363 . 2  |-  CC  e.  _V
2 absf 12825 . . 3  |-  abs : CC
--> RR
3 subf 9612 . . 3  |-  -  :
( CC  X.  CC )
--> CC
4 fco 5568 . . 3  |-  ( ( abs : CC --> RR  /\  -  : ( CC  X.  CC ) --> CC )  -> 
( abs  o.  -  ) : ( CC  X.  CC ) --> RR )
52, 3, 4mp2an 672 . 2  |-  ( abs 
o.  -  ) :
( CC  X.  CC )
--> RR
6 subcl 9609 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
76abs00ad 12779 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( abs `  (
x  -  y ) )  =  0  <->  (
x  -  y )  =  0 ) )
8 eqid 2443 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
98cnmetdval 20350 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x ( abs 
o.  -  ) y
)  =  ( abs `  ( x  -  y
) ) )
109eqcomd 2448 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( abs `  (
x  -  y ) )  =  ( x ( abs  o.  -  ) y ) )
1110eqeq1d 2451 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( abs `  (
x  -  y ) )  =  0  <->  (
x ( abs  o.  -  ) y )  =  0 ) )
12 subeq0 9635 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( x  -  y )  =  0  <-> 
x  =  y ) )
137, 11, 123bitr3d 283 . 2  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( x ( abs  o.  -  )
y )  =  0  <-> 
x  =  y ) )
14 abs3dif 12819 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  ( abs `  ( x  -  y ) )  <_ 
( ( abs `  (
x  -  z ) )  +  ( abs `  ( z  -  y
) ) ) )
15 abssub 12814 . . . . . 6  |-  ( ( x  e.  CC  /\  z  e.  CC )  ->  ( abs `  (
x  -  z ) )  =  ( abs `  ( z  -  x
) ) )
1615oveq1d 6106 . . . . 5  |-  ( ( x  e.  CC  /\  z  e.  CC )  ->  ( ( abs `  (
x  -  z ) )  +  ( abs `  ( z  -  y
) ) )  =  ( ( abs `  (
z  -  x ) )  +  ( abs `  ( z  -  y
) ) ) )
17163adant2 1007 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( abs `  (
x  -  z ) )  +  ( abs `  ( z  -  y
) ) )  =  ( ( abs `  (
z  -  x ) )  +  ( abs `  ( z  -  y
) ) ) )
1814, 17breqtrd 4316 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  ( abs `  ( x  -  y ) )  <_ 
( ( abs `  (
z  -  x ) )  +  ( abs `  ( z  -  y
) ) ) )
1993adant3 1008 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x ( abs  o.  -  ) y )  =  ( abs `  (
x  -  y ) ) )
208cnmetdval 20350 . . . . . 6  |-  ( ( z  e.  CC  /\  x  e.  CC )  ->  ( z ( abs 
o.  -  ) x
)  =  ( abs `  ( z  -  x
) ) )
21203adant3 1008 . . . . 5  |-  ( ( z  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
z ( abs  o.  -  ) x )  =  ( abs `  (
z  -  x ) ) )
228cnmetdval 20350 . . . . . 6  |-  ( ( z  e.  CC  /\  y  e.  CC )  ->  ( z ( abs 
o.  -  ) y
)  =  ( abs `  ( z  -  y
) ) )
23223adant2 1007 . . . . 5  |-  ( ( z  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
z ( abs  o.  -  ) y )  =  ( abs `  (
z  -  y ) ) )
2421, 23oveq12d 6109 . . . 4  |-  ( ( z  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
( z ( abs 
o.  -  ) x
)  +  ( z ( abs  o.  -  ) y ) )  =  ( ( abs `  ( z  -  x
) )  +  ( abs `  ( z  -  y ) ) ) )
25243coml 1194 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( z ( abs 
o.  -  ) x
)  +  ( z ( abs  o.  -  ) y ) )  =  ( ( abs `  ( z  -  x
) )  +  ( abs `  ( z  -  y ) ) ) )
2618, 19, 253brtr4d 4322 . 2  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x ( abs  o.  -  ) y )  <_  ( ( z ( abs  o.  -  ) x )  +  ( z ( abs 
o.  -  ) y
) ) )
271, 5, 13, 26ismeti 19900 1  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    X. cxp 4838    o. ccom 4844   -->wf 5414   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282    + caddc 9285    <_ cle 9419    - cmin 9595   abscabs 12723   Metcme 17802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-met 17811
This theorem is referenced by:  cnxmet  20352  cnfldms  20355  remet  20367  xrsdsre  20387  lebnumii  20538  cncmet  20833  cncms  20867  ovolctb  20973  dvlog2lem  22097  cnrrext  26439  cntotbnd  28695  iccbnd  28739  sblpnf  29596
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