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Theorem dscmet 21219
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
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0re 9613 . . . . . 6  |-  0  e.  RR
2 1re 9612 . . . . . 6  |-  1  e.  RR
31, 2keepel 4012 . . . . 5  |-  if ( x  =  y ,  0 ,  1 )  e.  RR
43rgen2w 2819 . . . 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 6866 . . . 4  |-  ( A. x  e.  X  A. y  e.  X  if ( x  =  y ,  0 ,  1 )  e.  RR  <->  D :
( X  X.  X
) --> RR )
74, 6mpbi 208 . . 3  |-  D :
( X  X.  X
) --> RR
8 equequ1 1799 . . . . . . . . 9  |-  ( x  =  w  ->  (
x  =  y  <->  w  =  y ) )
98ifbid 3966 . . . . . . . 8  |-  ( x  =  w  ->  if ( x  =  y ,  0 ,  1 )  =  if ( w  =  y ,  0 ,  1 ) )
10 equequ2 1800 . . . . . . . . 9  |-  ( y  =  v  ->  (
w  =  y  <->  w  =  v ) )
1110ifbid 3966 . . . . . . . 8  |-  ( y  =  v  ->  if ( w  =  y ,  0 ,  1 )  =  if ( w  =  v ,  0 ,  1 ) )
12 0nn0 10831 . . . . . . . . . 10  |-  0  e.  NN0
13 1nn0 10832 . . . . . . . . . 10  |-  1  e.  NN0
1412, 13keepel 4012 . . . . . . . . 9  |-  if ( w  =  v ,  0 ,  1 )  e.  NN0
1514elexi 3119 . . . . . . . 8  |-  if ( w  =  v ,  0 ,  1 )  e.  _V
169, 11, 5, 15ovmpt2 6437 . . . . . . 7  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( w D v )  =  if ( w  =  v ,  0 ,  1 ) )
1716eqeq1d 2459 . . . . . 6  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( ( w D v )  =  0  <-> 
if ( w  =  v ,  0 ,  1 )  =  0 ) )
18 iffalse 3953 . . . . . . . . . 10  |-  ( -.  w  =  v  ->  if ( w  =  v ,  0 ,  1 )  =  1 )
19 ax-1ne0 9578 . . . . . . . . . . 11  |-  1  =/=  0
2019a1i 11 . . . . . . . . . 10  |-  ( -.  w  =  v  -> 
1  =/=  0 )
2118, 20eqnetrd 2750 . . . . . . . . 9  |-  ( -.  w  =  v  ->  if ( w  =  v ,  0 ,  1 )  =/=  0 )
2221neneqd 2659 . . . . . . . 8  |-  ( -.  w  =  v  ->  -.  if ( w  =  v ,  0 ,  1 )  =  0 )
2322con4i 130 . . . . . . 7  |-  ( if ( w  =  v ,  0 ,  1 )  =  0  ->  w  =  v )
24 iftrue 3950 . . . . . . 7  |-  ( w  =  v  ->  if ( w  =  v ,  0 ,  1 )  =  0 )
2523, 24impbii 188 . . . . . 6  |-  ( if ( w  =  v ,  0 ,  1 )  =  0  <->  w  =  v )
2617, 25syl6bb 261 . . . . 5  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( ( w D v )  =  0  <-> 
w  =  v ) )
2712, 13keepel 4012 . . . . . . . . . . 11  |-  if ( u  =  w ,  0 ,  1 )  e.  NN0
2812, 13keepel 4012 . . . . . . . . . . 11  |-  if ( u  =  v ,  0 ,  1 )  e.  NN0
2927, 28nn0addcli 10854 . . . . . . . . . 10  |-  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN0
30 elnn0 10818 . . . . . . . . . 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 208 . . . . . . . . 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 4459 . . . . . . . . . . . 12  |-  ( 0  =  if ( w  =  v ,  0 ,  1 )  -> 
( 0  <_  1  <->  if ( w  =  v ,  0 ,  1 )  <_  1 ) )
33 breq1 4459 . . . . . . . . . . . 12  |-  ( 1  =  if ( w  =  v ,  0 ,  1 )  -> 
( 1  <_  1  <->  if ( w  =  v ,  0 ,  1 )  <_  1 ) )
34 0le1 10097 . . . . . . . . . . . 12  |-  0  <_  1
352leidi 10108 . . . . . . . . . . . 12  |-  1  <_  1
3632, 33, 34, 35keephyp 4009 . . . . . . . . . . 11  |-  if ( w  =  v ,  0 ,  1 )  <_  1
37 nnge1 10582 . . . . . . . . . . 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 10827 . . . . . . . . . . . 12  |-  if ( w  =  v ,  0 ,  1 )  e.  RR
3929nn0rei 10827 . . . . . . . . . . . 12  |-  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  RR
4038, 2, 39letri 9730 . . . . . . . . . . 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 663 . . . . . . . . . 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 10844 . . . . . . . . . . . . 13  |-  0  <_  if ( u  =  w ,  0 ,  1 )
4328nn0ge0i 10844 . . . . . . . . . . . . 13  |-  0  <_  if ( u  =  v ,  0 ,  1 )
4427nn0rei 10827 . . . . . . . . . . . . . 14  |-  if ( u  =  w ,  0 ,  1 )  e.  RR
4528nn0rei 10827 . . . . . . . . . . . . . 14  |-  if ( u  =  v ,  0 ,  1 )  e.  RR
4644, 45add20i 10117 . . . . . . . . . . . . 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 672 . . . . . . . . . . . 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 1800 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  w  ->  (
u  =  v  <->  u  =  w ) )
4948ifbid 3966 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  w  ->  if ( u  =  v ,  0 ,  1 )  =  if ( u  =  w ,  0 ,  1 ) )
5049eqeq1d 2459 . . . . . . . . . . . . . . . . 17  |-  ( v  =  w  ->  ( if ( u  =  v ,  0 ,  1 )  =  0  <->  if ( u  =  w ,  0 ,  1 )  =  0 ) )
5150, 48bibi12d 321 . . . . . . . . . . . . . . . 16  |-  ( v  =  w  ->  (
( if ( u  =  v ,  0 ,  1 )  =  0  <->  u  =  v
)  <->  ( if ( u  =  w ,  0 ,  1 )  =  0  <->  u  =  w ) ) )
52 equequ1 1799 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  =  u  ->  (
w  =  v  <->  u  =  v ) )
5352ifbid 3966 . . . . . . . . . . . . . . . . . . 19  |-  ( w  =  u  ->  if ( w  =  v ,  0 ,  1 )  =  if ( u  =  v ,  0 ,  1 ) )
5453eqeq1d 2459 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  u  ->  ( if ( w  =  v ,  0 ,  1 )  =  0  <->  if ( u  =  v ,  0 ,  1 )  =  0 ) )
5554, 52bibi12d 321 . . . . . . . . . . . . . . . . 17  |-  ( w  =  u  ->  (
( if ( w  =  v ,  0 ,  1 )  =  0  <->  w  =  v
)  <->  ( if ( u  =  v ,  0 ,  1 )  =  0  <->  u  =  v ) ) )
5655, 25chvarv 2015 . . . . . . . . . . . . . . . 16  |-  ( if ( u  =  v ,  0 ,  1 )  =  0  <->  u  =  v )
5751, 56chvarv 2015 . . . . . . . . . . . . . . 15  |-  ( if ( u  =  w ,  0 ,  1 )  =  0  <->  u  =  w )
58 eqtr2 2484 . . . . . . . . . . . . . . 15  |-  ( ( u  =  w  /\  u  =  v )  ->  w  =  v )
5957, 56, 58syl2anb 479 . . . . . . . . . . . . . 14  |-  ( ( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 )  ->  w  =  v )
6059iftrued 3952 . . . . . . . . . . . . 13  |-  ( ( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 )  ->  if ( w  =  v ,  0 ,  1 )  =  0 )
611leidi 10108 . . . . . . . . . . . . 13  |-  0  <_  0
6260, 61syl6eqbr 4493 . . . . . . . . . . . 12  |-  ( ( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 )  ->  if ( w  =  v ,  0 ,  1 )  <_  0 )
6347, 62sylbi 195 . . . . . . . . . . 11  |-  ( ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  =  0  ->  if ( w  =  v ,  0 ,  1 )  <_ 
0 )
64 id 22 . . . . . . . . . . 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 4482 . . . . . . . . . 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 379 . . . . . . . . 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 12 . . . . . . . 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 466 . . . . . . . 8  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
w D v )  =  if ( w  =  v ,  0 ,  1 ) )
69 eqeq12 2476 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  w )  ->  ( x  =  y  <-> 
u  =  w ) )
7069ifbid 3966 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  w )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( u  =  w ,  0 ,  1 ) )
7127elexi 3119 . . . . . . . . . . 11  |-  if ( u  =  w ,  0 ,  1 )  e.  _V
7270, 5, 71ovmpt2a 6432 . . . . . . . . . 10  |-  ( ( u  e.  X  /\  w  e.  X )  ->  ( u D w )  =  if ( u  =  w ,  0 ,  1 ) )
7372adantrr 716 . . . . . . . . 9  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
u D w )  =  if ( u  =  w ,  0 ,  1 ) )
74 eqeq12 2476 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  v )  ->  ( x  =  y  <-> 
u  =  v ) )
7574ifbid 3966 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  v )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( u  =  v ,  0 ,  1 ) )
7628elexi 3119 . . . . . . . . . . 11  |-  if ( u  =  v ,  0 ,  1 )  e.  _V
7775, 5, 76ovmpt2a 6432 . . . . . . . . . 10  |-  ( ( u  e.  X  /\  v  e.  X )  ->  ( u D v )  =  if ( u  =  v ,  0 ,  1 ) )
7877adantrl 715 . . . . . . . . 9  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
u D v )  =  if ( u  =  v ,  0 ,  1 ) )
7973, 78oveq12d 6314 . . . . . . . 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 4486 . . . . . . 7  |-  ( ( u  e.  X  /\  ( w  e.  X  /\  v  e.  X
) )  ->  (
w D v )  <_  ( ( u D w )  +  ( u D v ) ) )
8180expcom 435 . . . . . 6  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( u  e.  X  ->  ( w D v )  <_  ( (
u D w )  +  ( u D v ) ) ) )
8281ralrimiv 2869 . . . . 5  |-  ( ( w  e.  X  /\  v  e.  X )  ->  A. u  e.  X  ( w D v )  <_  ( (
u D w )  +  ( u D v ) ) )
8326, 82jca 532 . . . 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 2884 . . 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 455 . 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 20952 . 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 233 1  |-  ( X  e.  V  ->  D  e.  ( Met `  X
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   ifcif 3944   class class class wbr 4456    X. cxp 5006   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    <_ cle 9646   NNcn 10556   NN0cn0 10816   Metcme 18531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-met 18540
This theorem is referenced by:  dscopn  21220
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