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Theorem 0met 17946
Description: The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
0met  |-  (/)  e.  ( Met `  (/) )

Proof of Theorem 0met
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4166 . 2  |-  (/)  e.  _V
2 f0 5441 . . 3  |-  (/) : (/) --> RR
3 xp0 5114 . . . 4  |-  ( (/)  X.  (/) )  =  (/)
43feq2i 5400 . . 3  |-  ( (/) : ( (/)  X.  (/) ) --> RR  <->  (/) :
(/) --> RR )
52, 4mpbir 200 . 2  |-  (/) : (
(/)  X.  (/) ) --> RR
6 noel 3472 . . . 4  |-  -.  x  e.  (/)
76pm2.21i 123 . . 3  |-  ( x  e.  (/)  ->  ( (
x (/) y )  =  0  <->  x  =  y
) )
87adantr 451 . 2  |-  ( ( x  e.  (/)  /\  y  e.  (/) )  ->  (
( x (/) y )  =  0  <->  x  =  y ) )
96pm2.21i 123 . . 3  |-  ( x  e.  (/)  ->  ( x (/) y )  <_  (
( z (/) x )  +  ( z (/) y ) ) )
1093ad2ant1 976 . 2  |-  ( ( x  e.  (/)  /\  y  e.  (/)  /\  z  e.  (/) )  ->  ( x
(/) y )  <_ 
( ( z (/) x )  +  ( z (/) y ) ) )
111, 5, 8, 10ismeti 17906 1  |-  (/)  e.  ( Met `  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   (/)c0 3468   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    + caddc 8756    <_ cle 8884   Metcme 16386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-met 16390
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