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Theorem dscmet 18022
Description: The discrete metric on any set  X. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
Hypothesis
Ref Expression
dscmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( x  =  y ,  0 ,  1 ) )
Assertion
Ref Expression
dscmet  |-  ( X  e.  V  ->  D  e.  ( Met `  X
) )
Distinct variable group:    x, y, X
Allowed substitution hints:    D( x, y)    V( x, y)

Proof of Theorem dscmet
StepHypRef Expression
1 0re 8771 . . . . . 6  |-  0  e.  RR
2 1re 8770 . . . . . 6  |-  1  e.  RR
31, 2keepel 3563 . . . . 5  |-  if ( x  =  y ,  0 ,  1 )  e.  RR
43rgen2w 2582 . . . 4  |-  A. x  e.  X  A. y  e.  X  if (
x  =  y ,  0 ,  1 )  e.  RR
5 dscmet.1 . . . . 5  |-  D  =  ( x  e.  X ,  y  e.  X  |->  if ( x  =  y ,  0 ,  1 ) )
65fmpt2 6090 . . . 4  |-  ( A. x  e.  X  A. y  e.  X  if ( x  =  y ,  0 ,  1 )  e.  RR  <->  D :
( X  X.  X
) --> RR )
74, 6mpbi 201 . . 3  |-  D :
( X  X.  X
) --> RR
8 equequ1 1829 . . . . . . . . 9  |-  ( x  =  w  ->  (
x  =  y  <->  w  =  y ) )
98ifbid 3524 . . . . . . . 8  |-  ( x  =  w  ->  if ( x  =  y ,  0 ,  1 )  =  if ( w  =  y ,  0 ,  1 ) )
10 equequ2 1830 . . . . . . . . 9  |-  ( y  =  v  ->  (
w  =  y  <->  w  =  v ) )
1110ifbid 3524 . . . . . . . 8  |-  ( y  =  v  ->  if ( w  =  y ,  0 ,  1 )  =  if ( w  =  v ,  0 ,  1 ) )
12 0nn0 9912 . . . . . . . . . 10  |-  0  e.  NN0
13 1nn0 9913 . . . . . . . . . 10  |-  1  e.  NN0
1412, 13keepel 3563 . . . . . . . . 9  |-  if ( w  =  v ,  0 ,  1 )  e.  NN0
1514elexi 2749 . . . . . . . 8  |-  if ( w  =  v ,  0 ,  1 )  e.  _V
169, 11, 5, 15ovmpt2 5882 . . . . . . 7  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( w D v )  =  if ( w  =  v ,  0 ,  1 ) )
1716eqeq1d 2264 . . . . . 6  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( ( w D v )  =  0  <-> 
if ( w  =  v ,  0 ,  1 )  =  0 ) )
18 iffalse 3513 . . . . . . . . . 10  |-  ( -.  w  =  v  ->  if ( w  =  v ,  0 ,  1 )  =  1 )
19 ax-1ne0 8739 . . . . . . . . . . 11  |-  1  =/=  0
2019a1i 12 . . . . . . . . . 10  |-  ( -.  w  =  v  -> 
1  =/=  0 )
2118, 20eqnetrd 2437 . . . . . . . . 9  |-  ( -.  w  =  v  ->  if ( w  =  v ,  0 ,  1 )  =/=  0 )
2221neneqd 2435 . . . . . . . 8  |-  ( -.  w  =  v  ->  -.  if ( w  =  v ,  0 ,  1 )  =  0 )
2322con4i 124 . . . . . . 7  |-  ( if ( w  =  v ,  0 ,  1 )  =  0  ->  w  =  v )
24 iftrue 3512 . . . . . . 7  |-  ( w  =  v  ->  if ( w  =  v ,  0 ,  1 )  =  0 )
2523, 24impbii 182 . . . . . 6  |-  ( if ( w  =  v ,  0 ,  1 )  =  0  <->  w  =  v )
2617, 25syl6bb 254 . . . . 5  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( ( w D v )  =  0  <-> 
w  =  v ) )
2712, 13keepel 3563 . . . . . . . . . . 11  |-  if ( u  =  w ,  0 ,  1 )  e.  NN0
2812, 13keepel 3563 . . . . . . . . . . 11  |-  if ( u  =  v ,  0 ,  1 )  e.  NN0
2927, 28nn0addcli 9933 . . . . . . . . . 10  |-  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN0
30 elnn0 9899 . . . . . . . . . 10  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN0  <->  (
( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN  \/  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0 ) )
3129, 30mpbi 201 . . . . . . . . 9  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN  \/  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0 )
32 breq1 3966 . . . . . . . . . . . 12  |-  ( 0  =  if ( w  =  v ,  0 ,  1 )  -> 
( 0  <_  1  <->  if ( w  =  v ,  0 ,  1 )  <_  1 ) )
33 breq1 3966 . . . . . . . . . . . 12  |-  ( 1  =  if ( w  =  v ,  0 ,  1 )  -> 
( 1  <_  1  <->  if ( w  =  v ,  0 ,  1 )  <_  1 ) )
34 0le1 9230 . . . . . . . . . . . 12  |-  0  <_  1
352leidi 9240 . . . . . . . . . . . 12  |-  1  <_  1
3632, 33, 34, 35keephyp 3560 . . . . . . . . . . 11  |-  if ( w  =  v ,  0 ,  1 )  <_  1
37 nnge1 9705 . . . . . . . . . . 11  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN  ->  1  <_  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) ) )
3814nn0rei 9908 . . . . . . . . . . . 12  |-  if ( w  =  v ,  0 ,  1 )  e.  RR
3929nn0rei 9908 . . . . . . . . . . . 12  |-  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  RR
4038, 2, 39letri 8881 . . . . . . . . . . 11  |-  ( ( if ( w  =  v ,  0 ,  1 )  <_  1  /\  1  <_  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) ) )  ->  if ( w  =  v ,  0 ,  1 )  <_  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) ) )
4136, 37, 40sylancr 647 . . . . . . . . . 10  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN  ->  if ( w  =  v ,  0 ,  1 )  <_  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) ) )
4227nn0ge0i 9925 . . . . . . . . . . . . 13  |-  0  <_  if ( u  =  w ,  0 ,  1 )
4328nn0ge0i 9925 . . . . . . . . . . . . 13  |-  0  <_  if ( u  =  v ,  0 ,  1 )
4427nn0rei 9908 . . . . . . . . . . . . . 14  |-  if ( u  =  w ,  0 ,  1 )  e.  RR
4528nn0rei 9908 . . . . . . . . . . . . . 14  |-  if ( u  =  v ,  0 ,  1 )  e.  RR
4644, 45add20i 9249 . . . . . . . . . . . . 13  |-  ( ( 0  <_  if (
u  =  w ,  0 ,  1 )  /\  0  <_  if ( u  =  v ,  0 ,  1 ) )  ->  (
( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0  <->  ( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 ) ) )
4742, 43, 46mp2an 656 . . . . . . . . . . . 12  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0  <-> 
( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 ) )
48 equequ2 1830 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  w  ->  (
u  =  v  <->  u  =  w ) )
4948ifbid 3524 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  w  ->  if ( u  =  v ,  0 ,  1 )  =  if ( u  =  w ,  0 ,  1 ) )
5049eqeq1d 2264 . . . . . . . . . . . . . . . . 17  |-  ( v  =  w  ->  ( if ( u  =  v ,  0 ,  1 )  =  0  <->  if ( u  =  w ,  0 ,  1 )  =  0 ) )
5150, 48bibi12d 314 . . . . . . . . . . . . . . . 16  |-  ( v  =  w  ->  (
( if ( u  =  v ,  0 ,  1 )  =  0  <->  u  =  v
)  <->  ( if ( u  =  w ,  0 ,  1 )  =  0  <->  u  =  w ) ) )
52 equequ1 1829 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  =  u  ->  (
w  =  v  <->  u  =  v ) )
5352ifbid 3524 . . . . . . . . . . . . . . . . . . 19  |-  ( w  =  u  ->  if ( w  =  v ,  0 ,  1 )  =  if ( u  =  v ,  0 ,  1 ) )
5453eqeq1d 2264 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  u  ->  ( if ( w  =  v ,  0 ,  1 )  =  0  <->  if ( u  =  v ,  0 ,  1 )  =  0 ) )
5554, 52bibi12d 314 . . . . . . . . . . . . . . . . 17  |-  ( w  =  u  ->  (
( if ( w  =  v ,  0 ,  1 )  =  0  <->  w  =  v
)  <->  ( if ( u  =  v ,  0 ,  1 )  =  0  <->  u  =  v ) ) )
5655, 25chvarv 2062 . . . . . . . . . . . . . . . 16  |-  ( if ( u  =  v ,  0 ,  1 )  =  0  <->  u  =  v )
5751, 56chvarv 2062 . . . . . . . . . . . . . . 15  |-  ( if ( u  =  w ,  0 ,  1 )  =  0  <->  u  =  w )
58 eqtr2 2274 . . . . . . . . . . . . . . 15  |-  ( ( u  =  w  /\  u  =  v )  ->  w  =  v )
5957, 56, 58syl2anb 467 . . . . . . . . . . . . . 14  |-  ( ( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 )  ->  w  =  v )
6059, 24syl 17 . . . . . . . . . . . . 13  |-  ( ( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 )  ->  if ( w  =  v ,  0 ,  1 )  =  0 )
611leidi 9240 . . . . . . . . . . . . 13  |-  0  <_  0
6260, 61syl6eqbr 4000 . . . . . . . . . . . 12  |-  ( ( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 )  ->  if ( w  =  v ,  0 ,  1 )  <_  0 )
6347, 62sylbi 189 . . . . . . . . . . 11  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0  ->  if ( w  =  v ,  0 ,  1 )  <_ 
0 )
64 id 21 . . . . . . . . . . 11  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0  ->  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0 )
6563, 64breqtrrd 3989 . . . . . . . . . 10  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0  ->  if ( w  =  v ,  0 ,  1 )  <_ 
( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) ) )
6641, 65jaoi 370 . . . . . . . . 9  |-  ( ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN  \/  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0 )  ->  if ( w  =  v ,  0 ,  1 )  <_ 
( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) ) )
6731, 66mp1i 13 . . . . . . . 8  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  if ( w  =  v ,  0 ,  1 )  <_  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) ) )
6816adantl 454 . . . . . . . 8  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
w D v )  =  if ( w  =  v ,  0 ,  1 ) )
69 eqeq12 2268 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  w )  ->  ( x  =  y  <-> 
u  =  w ) )
7069ifbid 3524 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  w )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( u  =  w ,  0 ,  1 ) )
7127elexi 2749 . . . . . . . . . . 11  |-  if ( u  =  w ,  0 ,  1 )  e.  _V
7270, 5, 71ovmpt2a 5877 . . . . . . . . . 10  |-  ( ( u  e.  X  /\  w  e.  X )  ->  ( u D w )  =  if ( u  =  w ,  0 ,  1 ) )
7372adantrr 700 . . . . . . . . 9  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
u D w )  =  if ( u  =  w ,  0 ,  1 ) )
74 eqeq12 2268 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  v )  ->  ( x  =  y  <-> 
u  =  v ) )
7574ifbid 3524 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  v )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( u  =  v ,  0 ,  1 ) )
7628elexi 2749 . . . . . . . . . . 11  |-  if ( u  =  v ,  0 ,  1 )  e.  _V
7775, 5, 76ovmpt2a 5877 . . . . . . . . . 10  |-  ( ( u  e.  X  /\  v  e.  X )  ->  ( u D v )  =  if ( u  =  v ,  0 ,  1 ) )
7877adantrl 699 . . . . . . . . 9  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
u D v )  =  if ( u  =  v ,  0 ,  1 ) )
7973, 78oveq12d 5775 . . . . . . . 8  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
( u D w )  +  ( u D v ) )  =  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) ) )
8067, 68, 793brtr4d 3993 . . . . . . 7  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
w D v )  <_  ( ( u D w )  +  ( u D v ) ) )
8180expcom 426 . . . . . 6  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( u  e.  X  ->  ( w D v )  <_  ( (
u D w )  +  ( u D v ) ) ) )
8281ralrimiv 2596 . . . . 5  |-  ( ( w  e.  X  /\  v  e.  X )  ->  A. u  e.  X  ( w D v )  <_  ( (
u D w )  +  ( u D v ) ) )
8326, 82jca 520 . . . 4  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( ( ( w D v )  =  0  <->  w  =  v
)  /\  A. u  e.  X  ( w D v )  <_ 
( ( u D w )  +  ( u D v ) ) ) )
8483rgen2a 2580 . . 3  |-  A. w  e.  X  A. v  e.  X  ( (
( w D v )  =  0  <->  w  =  v )  /\  A. u  e.  X  ( w D v )  <_  ( ( u D w )  +  ( u D v ) ) )
857, 84pm3.2i 443 . 2  |-  ( D : ( X  X.  X ) --> RR  /\  A. w  e.  X  A. v  e.  X  (
( ( w D v )  =  0  <-> 
w  =  v )  /\  A. u  e.  X  ( w D v )  <_  (
( u D w )  +  ( u D v ) ) ) )
86 ismet 17815 . 2  |-  ( X  e.  V  ->  ( D  e.  ( Met `  X )  <->  ( D : ( X  X.  X ) --> RR  /\  A. w  e.  X  A. v  e.  X  (
( ( w D v )  =  0  <-> 
w  =  v )  /\  A. u  e.  X  ( w D v )  <_  (
( u D w )  +  ( u D v ) ) ) ) ) )
8785, 86mpbiri 226 1  |-  ( X  e.  V  ->  D  e.  ( Met `  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   ifcif 3506   class class class wbr 3963    X. cxp 4624   -->wf 4634   ` cfv 4638  (class class class)co 5757    e. cmpt2 5759   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    <_ cle 8801   NNcn 9679   NN0cn0 9897   Metcme 16297
This theorem is referenced by:  dscopn  18023
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-n0 9898  df-met 16301
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