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Theorem cnmet 21007
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 9562 . 2  |-  CC  e.  _V
2 absf 13119 . . 3  |-  abs : CC
--> RR
3 subf 9811 . . 3  |-  -  :
( CC  X.  CC )
--> CC
4 fco 5732 . . 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 9808 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  -  y
)  e.  CC )
76abs00ad 13073 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( abs `  (
x  -  y ) )  =  0  <->  (
x  -  y )  =  0 ) )
8 eqid 2460 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
98cnmetdval 21006 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x ( abs 
o.  -  ) y
)  =  ( abs `  ( x  -  y
) ) )
109eqcomd 2468 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( abs `  (
x  -  y ) )  =  ( x ( abs  o.  -  ) y ) )
1110eqeq1d 2462 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( abs `  (
x  -  y ) )  =  0  <->  (
x ( abs  o.  -  ) y )  =  0 ) )
12 subeq0 9834 . . 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 13113 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  ( abs `  ( x  -  y ) )  <_ 
( ( abs `  (
x  -  z ) )  +  ( abs `  ( z  -  y
) ) ) )
15 abssub 13108 . . . . . 6  |-  ( ( x  e.  CC  /\  z  e.  CC )  ->  ( abs `  (
x  -  z ) )  =  ( abs `  ( z  -  x
) ) )
1615oveq1d 6290 . . . . 5  |-  ( ( x  e.  CC  /\  z  e.  CC )  ->  ( ( abs `  (
x  -  z ) )  +  ( abs `  ( z  -  y
) ) )  =  ( ( abs `  (
z  -  x ) )  +  ( abs `  ( z  -  y
) ) ) )
17163adant2 1010 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
( abs `  (
x  -  z ) )  +  ( abs `  ( z  -  y
) ) )  =  ( ( abs `  (
z  -  x ) )  +  ( abs `  ( z  -  y
) ) ) )
1814, 17breqtrd 4464 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  ( abs `  ( x  -  y ) )  <_ 
( ( abs `  (
z  -  x ) )  +  ( abs `  ( z  -  y
) ) ) )
1993adant3 1011 . . 3  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x ( abs  o.  -  ) y )  =  ( abs `  (
x  -  y ) ) )
208cnmetdval 21006 . . . . . 6  |-  ( ( z  e.  CC  /\  x  e.  CC )  ->  ( z ( abs 
o.  -  ) x
)  =  ( abs `  ( z  -  x
) ) )
21203adant3 1011 . . . . 5  |-  ( ( z  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
z ( abs  o.  -  ) x )  =  ( abs `  (
z  -  x ) ) )
228cnmetdval 21006 . . . . . 6  |-  ( ( z  e.  CC  /\  y  e.  CC )  ->  ( z ( abs 
o.  -  ) y
)  =  ( abs `  ( z  -  y
) ) )
23223adant2 1010 . . . . 5  |-  ( ( z  e.  CC  /\  x  e.  CC  /\  y  e.  CC )  ->  (
z ( abs  o.  -  ) y )  =  ( abs `  (
z  -  y ) ) )
2421, 23oveq12d 6293 . . . 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 1198 . . 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 4470 . 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 20556 1  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    X. cxp 4990    o. ccom 4996   -->wf 5575   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484    <_ cle 9618    - cmin 9794   abscabs 13017   Metcme 18168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-met 18177
This theorem is referenced by:  cnxmet  21008  cnfldms  21011  remet  21023  xrsdsre  21043  lebnumii  21194  cncmet  21489  cncms  21523  ovolctb  21629  dvlog2lem  22754  cnrrext  27613  cntotbnd  29882  iccbnd  29926  sblpnf  30782
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