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Theorem dscmet 20168
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 9389 . . . . . 6  |-  0  e.  RR
2 1re 9388 . . . . . 6  |-  1  e.  RR
31, 2keepel 3860 . . . . 5  |-  if ( x  =  y ,  0 ,  1 )  e.  RR
43rgen2w 2787 . . . 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 6644 . . . 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 1736 . . . . . . . . 9  |-  ( x  =  w  ->  (
x  =  y  <->  w  =  y ) )
98ifbid 3814 . . . . . . . 8  |-  ( x  =  w  ->  if ( x  =  y ,  0 ,  1 )  =  if ( w  =  y ,  0 ,  1 ) )
10 equequ2 1737 . . . . . . . . 9  |-  ( y  =  v  ->  (
w  =  y  <->  w  =  v ) )
1110ifbid 3814 . . . . . . . 8  |-  ( y  =  v  ->  if ( w  =  y ,  0 ,  1 )  =  if ( w  =  v ,  0 ,  1 ) )
12 0nn0 10597 . . . . . . . . . 10  |-  0  e.  NN0
13 1nn0 10598 . . . . . . . . . 10  |-  1  e.  NN0
1412, 13keepel 3860 . . . . . . . . 9  |-  if ( w  =  v ,  0 ,  1 )  e.  NN0
1514elexi 2985 . . . . . . . 8  |-  if ( w  =  v ,  0 ,  1 )  e.  _V
169, 11, 5, 15ovmpt2 6229 . . . . . . 7  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( w D v )  =  if ( w  =  v ,  0 ,  1 ) )
1716eqeq1d 2451 . . . . . 6  |-  ( ( w  e.  X  /\  v  e.  X )  ->  ( ( w D v )  =  0  <-> 
if ( w  =  v ,  0 ,  1 )  =  0 ) )
18 iffalse 3802 . . . . . . . . . 10  |-  ( -.  w  =  v  ->  if ( w  =  v ,  0 ,  1 )  =  1 )
19 ax-1ne0 9354 . . . . . . . . . . 11  |-  1  =/=  0
2019a1i 11 . . . . . . . . . 10  |-  ( -.  w  =  v  -> 
1  =/=  0 )
2118, 20eqnetrd 2629 . . . . . . . . 9  |-  ( -.  w  =  v  ->  if ( w  =  v ,  0 ,  1 )  =/=  0 )
2221neneqd 2627 . . . . . . . 8  |-  ( -.  w  =  v  ->  -.  if ( w  =  v ,  0 ,  1 )  =  0 )
2322con4i 130 . . . . . . 7  |-  ( if ( w  =  v ,  0 ,  1 )  =  0  ->  w  =  v )
24 iftrue 3800 . . . . . . 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 3860 . . . . . . . . . . 11  |-  if ( u  =  w ,  0 ,  1 )  e.  NN0
2812, 13keepel 3860 . . . . . . . . . . 11  |-  if ( u  =  v ,  0 ,  1 )  e.  NN0
2927, 28nn0addcli 10620 . . . . . . . . . 10  |-  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  NN0
30 elnn0 10584 . . . . . . . . . 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 4298 . . . . . . . . . . . 12  |-  ( 0  =  if ( w  =  v ,  0 ,  1 )  -> 
( 0  <_  1  <->  if ( w  =  v ,  0 ,  1 )  <_  1 ) )
33 breq1 4298 . . . . . . . . . . . 12  |-  ( 1  =  if ( w  =  v ,  0 ,  1 )  -> 
( 1  <_  1  <->  if ( w  =  v ,  0 ,  1 )  <_  1 ) )
34 0le1 9866 . . . . . . . . . . . 12  |-  0  <_  1
352leidi 9877 . . . . . . . . . . . 12  |-  1  <_  1
3632, 33, 34, 35keephyp 3857 . . . . . . . . . . 11  |-  if ( w  =  v ,  0 ,  1 )  <_  1
37 nnge1 10351 . . . . . . . . . . 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 10593 . . . . . . . . . . . 12  |-  if ( w  =  v ,  0 ,  1 )  e.  RR
3929nn0rei 10593 . . . . . . . . . . . 12  |-  ( if ( u  =  w ,  0 ,  1 )  +  if ( u  =  v ,  0 ,  1 ) )  e.  RR
4038, 2, 39letri 9506 . . . . . . . . . . 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 10610 . . . . . . . . . . . . 13  |-  0  <_  if ( u  =  w ,  0 ,  1 )
4328nn0ge0i 10610 . . . . . . . . . . . . 13  |-  0  <_  if ( u  =  v ,  0 ,  1 )
4427nn0rei 10593 . . . . . . . . . . . . . 14  |-  if ( u  =  w ,  0 ,  1 )  e.  RR
4528nn0rei 10593 . . . . . . . . . . . . . 14  |-  if ( u  =  v ,  0 ,  1 )  e.  RR
4644, 45add20i 9886 . . . . . . . . . . . . 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 1737 . . . . . . . . . . . . . . . . . . 19  |-  ( v  =  w  ->  (
u  =  v  <->  u  =  w ) )
4948ifbid 3814 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  w  ->  if ( u  =  v ,  0 ,  1 )  =  if ( u  =  w ,  0 ,  1 ) )
5049eqeq1d 2451 . . . . . . . . . . . . . . . . 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 1736 . . . . . . . . . . . . . . . . . . . 20  |-  ( w  =  u  ->  (
w  =  v  <->  u  =  v ) )
5352ifbid 3814 . . . . . . . . . . . . . . . . . . 19  |-  ( w  =  u  ->  if ( w  =  v ,  0 ,  1 )  =  if ( u  =  v ,  0 ,  1 ) )
5453eqeq1d 2451 . . . . . . . . . . . . . . . . . 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 1958 . . . . . . . . . . . . . . . 16  |-  ( if ( u  =  v ,  0 ,  1 )  =  0  <->  u  =  v )
5751, 56chvarv 1958 . . . . . . . . . . . . . . 15  |-  ( if ( u  =  w ,  0 ,  1 )  =  0  <->  u  =  w )
58 eqtr2 2461 . . . . . . . . . . . . . . 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 )
6059, 24syl 16 . . . . . . . . . . . . 13  |-  ( ( if ( u  =  w ,  0 ,  1 )  =  0  /\  if ( u  =  v ,  0 ,  1 )  =  0 )  ->  if ( w  =  v ,  0 ,  1 )  =  0 )
611leidi 9877 . . . . . . . . . . . . 13  |-  0  <_  0
6260, 61syl6eqbr 4332 . . . . . . . . . . . 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 4321 . . . . . . . . . 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 2455 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  w )  ->  ( x  =  y  <-> 
u  =  w ) )
7069ifbid 3814 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  w )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( u  =  w ,  0 ,  1 ) )
7127elexi 2985 . . . . . . . . . . 11  |-  if ( u  =  w ,  0 ,  1 )  e.  _V
7270, 5, 71ovmpt2a 6224 . . . . . . . . . 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 2455 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  v )  ->  ( x  =  y  <-> 
u  =  v ) )
7574ifbid 3814 . . . . . . . . . . 11  |-  ( ( x  =  u  /\  y  =  v )  ->  if ( x  =  y ,  0 ,  1 )  =  if ( u  =  v ,  0 ,  1 ) )
7628elexi 2985 . . . . . . . . . . 11  |-  if ( u  =  v ,  0 ,  1 )  e.  _V
7775, 5, 76ovmpt2a 6224 . . . . . . . . . 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 6112 . . . . . . . 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 4325 . . . . . . 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 2801 . . . . 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 2785 . . 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 19901 . 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 1369    e. wcel 1756    =/= wne 2609   A.wral 2718   ifcif 3794   class class class wbr 4295    X. cxp 4841   -->wf 5417   ` cfv 5421  (class class class)co 6094    e. cmpt2 6096   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288    <_ cle 9422   NNcn 10325   NN0cn0 10582   Metcme 17805
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 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
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 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-n0 10583  df-met 17814
This theorem is referenced by:  dscopn  20169
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